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Introduction and guidance

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

This free badged course, Everyday maths 2, is an introduction to Level 2 Essential Skills in maths. It is designed to inspire you to improve your current maths skills and help you to remember any areas that you may have forgotten. Working through the examples and interactive activities in this course will help you to, among other things, run a household or make progress in your career.

You can work through the course at your own pace. To complete the course you will need access to a calculator and a notepad and pen.

The course has four sessions, with a total study time of approximately 48 hours. The sessions cover the following topics: numbers, measurement, shapes and space, and data. There will be plenty of examples to help you as you progress, together with opportunities to practise your understanding.

The regular interactive quizzes form part of this practice, and the end-of-course quiz is an opportunity to earn a badge that demonstrates your new skills. You can read more on how to study the course and about badges in the next sections.

After completing this course you will be able to:

understand practical problems, some of which are non-routine
identify the maths skills you need to tackle a problem
use maths in an organised way to find the solution you’re looking for
use appropriate checking procedures at each stage
explain the process you used to get an answer and draw simple conclusions from it.

Moving around the course

The easiest way to navigate around the course is through the ‘My course progress’ page. You can get back there at any time by clicking on ‘Back to course’ in the menu bar.

It’s also good practice, if you access a link from within a course page (including links to the quizzes), to open it in a new window or tab. That way you can easily return to where you’ve come from without having to use the back button on your browser.

What is a badged course?

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

While studying Everyday maths 2 you have the option to work towards gaining a digital badge.

Badged courses are a key part of The Open University’s mission to promote the educational well-being of the community. The courses also provide another way of helping you to progress from informal to formal learning.

To complete a course you need to be able to find about 48 hours of study time. It is possible to study them at any time, and at a pace to suit you.

Badged courses are all available on The Open University’s OpenLearn website and do not cost anything to study. They differ from Open University courses because you do not receive support from a tutor. But you do get useful feedback from the interactive quizzes.

What is a badge?

Digital badges are a new way of demonstrating online that you have gained a skill. Schools, colleges and universities are working with employers and other organisations to develop open badges that help learners gain recognition for their skills, and support employers to identify the right candidate for a job.

Badges demonstrate your work and achievement on the course. You can share your achievement with friends, family and employers, and on social media. Badges are a great motivation, helping you to reach the end of the course. Gaining a badge often boosts confidence in the skills and abilities that underpin successful study. So, completing this course should encourage you to think about taking other courses, for example enrolling at a college for a formal qualification. (You will be given details on this at the end of the course.)

How to get a badge

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Getting a badge is straightforward! Here’s what you have to do:

read all of the pages of the course
score 70% or more in the end-of-course quiz.

For all the quizzes, you can have three attempts at most of the questions (for true or false type questions you usually only get one attempt). If you get the answer right first time you will get more marks than for a correct answer the second or third time. Therefore, please be aware that for the end-of-course quiz it is possible to get all the questions right but not score 50% and be eligible for the OpenLearn badge on that attempt. If one of your answers is incorrect you will often receive helpful feedback and suggestions about how to work out the correct answer.

If you’re not successful in getting 70% in the end-of-course quiz the first time, after 24 hours you can attempt it again and come back as many times as you like.

We hope that as many people as possible will gain an Open University badge – so you should see getting a badge as an opportunity to reflect on what you have learned rather than as a test.

If you need more guidance on getting a badge and what you can do with it, take a look at the OpenLearn FAQs. When you gain your badge you will receive an email to notify you and you will be able to view and manage all your badges in My OpenLearn within 24 hours of completing the criteria to gain a badge.

Now get started with Session 1.

Introduction

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

With an understanding of numbers you have the basic skills you need to deal with much of everyday life. In this session, you will learn about how to use each of the four different mathematical operations and how they apply to real life situations. Then you will encounter fractions and percentages which are incredibly useful when trying to work out the price of an item on special offer. Additionally, you will learn how to use ratio – handy if you are doing any baking – and how to work with formulas.

Throughout the course there will be various activities for you to complete in order to check your skills. All activities come with answers and show suggested working. It is important to know that there are often many ways of working out the same calculation. If your working is different to that shown but you arrive at the same answer, that’s perfectly fine. If you need to refresh your written methods for addition, subtraction, multiplication or division please refer to the relevant number section in Everyday maths 1.

If you want more practice with your maths skills, please contact your local college to discuss the options available.

By the end of this session you will be able to:

use the four operations to solve problems in context
understand rounding and look at different ways of doing this
write large numbers in full and shortened forms
carry out calculations with large numbers
carry out multistage calculations
solve problems involving negative numbers
define some key mathematical terms (multiple, lowest common multiple, factor, common factor and prime number)
identify lowest common multiples and factors
use fractions, decimals and percentages and convert between them
solve different types of ratio problems
make substitutions within given formulas to solve problems
use inverse operations and estimations to check your calculations.

Download this video clip.Video player: bltl_a69_numbers.mp4

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Transcript

[MUSIC PLAYING]

Even for something as basic as food shopping, developing your knowledge of numbers can be a real benefit, especially if you're looking for a bargain. Like, learning when to round a number up or down to help work out a rough estimate of what you need to buy. And understanding things like percentages can help you track exactly what bargains there are on offer. And when there is a chance to save money.

If you use a credit card, understanding interest rates will help you know how much it costs to use it. And when you come to cooking, getting to grips with formulas will help you when working out a cooking time. Or calculating how much money you have available, in case of an emergency.

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1 Four operations

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

You will already be using the four operations in your daily life (whether you realise it or not). Everyday life requires you to carry out maths all the time; checking you’ve been given the correct change, working out how many packs of cakes you need for the children’s birthday party and splitting the bill in a restaurant are all examples that come to mind.

Figure _unit2.1.1 Figure 1 Fruit maths puzzle

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Four maths sums illustrated with pieces of fruit. The first line shows: apple plus apple plus apple equals 30. The second line shows: apple plus 4 bananas plus 4 bananas equals 18. The third line shows: 4 bananas minus 2 coconuts equals 2. The fourth line shows: coconut plus apple plus 3 bananas equals ‘?’ (a question mark).

Figure 1 Fruit maths puzzle

The four operations are addition, subtraction, multiplication and division. You need to understand what each operation does and when to use it and at level 2, you will often be required to use more than one operation to answer a question.

Addition (+)
- This operation is used when you want to find the total, or sum, of two or more amounts.
Subtraction (−)
- This operation is used when you want to find the difference between two amounts or how much of something you have left after a quantity is used – for example, if you want to find the change owed after spending an amount of money.
Multiplication (×)
- This operation is used to calculate multiple amounts of the same number. For example, if you wanted to know the cost of 16 plants costing £4.75 each you would multiply £4.75 by 16:
  £4.75 × 16 = £76
Division (÷)
- Division is used when sharing or grouping items. For example, if a group of 6 friends won £765 on the lottery and you wanted to know how much each person would get you would divide £765 by 6:
  £765 ÷ 6 = £127.50

If you are doing this course to prepare you for Essential Skills Wales Application of Number, remember the exam papers do not allow the use of a calculator so please attempt the calculations in this session by hand.

Activity _unit2.1.1 Activity 1: Operation choice

Each of the four questions below uses one of the four operations. Match the operation to the question.

Using the following two lists, match each numbered item with the correct letter.

a.You go to the local café and buy a coffee for £2.35, a tea for £1.40 and a croissant for £1.85. How much do you spend?
b.Fourteen members of the same family go on holiday together. They each pay £155. What is the total cost of the holiday?
c.You make an insurance claim worth £18 950. The insurance company pays you £12 648. What is the difference between what you claimed and what you actually received?
d.You need to save £306 for a holiday. You have 18 months to save up that much money. How much do you need to save per month?

The correct answers are:

1 = d
2 = b
3 = c
4 = a

Try these questions to make sure that you are confident using each of the four operations.

Activity _unit2.1.2 Activity 2: Calculations with whole numbers

1245 + 654
187 + 65 401
1060 − 264
2000 − 173
543 × 19
1732 × 46
1312 ÷ 8
1044 ÷ 12
Stuart is saving £225 per month for a new car. How much would he have saved in:
- a.1 year
- b.2 years
A concert venue has 2190 seats arranged into 15 sections. How many seats are in each section?
A college sells 325 tickets to the charity ball at £12 each. How much money do they make from ticket sales?

Answer

1245 + 654 = 1899
187 + 65 401 = 65 588
1060 − 264 = 796
2000 − 173 = 1827
543 × 19 = 10 317
1732 × 46 = 79 672
1312 ÷ 8 = 164
1044 ÷ 12 = 87
- a.Stuart is saving £225 per month for 1 year or 12 months so the calculation is 225 × 12 = £2700.
- b.Stuart saves £2700 in 1 year so to work out the amount saved in 2 years the calculation is 2 × £2700 = £5400. Alternatively, you could add £2700 + £2700 = £5400.
To calculate this you divide the number of seats by the number of sections.
So, 2190 ÷ 15 = 146. So there are 146 seats in each section.
To calculate this you multiply the number of tickets by the cost per ticket.
So, 325 × 12 = 3900. So the college will make £3900 from ticket sales.

1.1 Expressing a remainder as a decimal

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

To split a prize of £125 between 5 friends you would do this calculation:

£125 ÷ 5 and get the answer £25.

This is a convenient, exact amount of money. However, often when you perform calculations, especially those involving division, you do not always get an answer that is suitable for the question.

For example, if there were 4 friends who shared the same prize we would do the calculation £125 ÷ 4 and get the answer £31 remainder £1. If we did the same calculation on a calculator you would get the answer £31.25, the remainder has been converted into a decimal. Let’s look at how to express the remainder as a decimal.

You can write one hundred and twenty five pounds in two different ways: £125 or £125.00. Both ways show the same amount but the second way allows you to continue the calculation and express it as a decimal.

Figure _unit2.1.2 Figure 2 Expressed as a decimal: 125 ÷ 4

We can use the same principal with any whole number, adding as many zeros after the decimal point as required. Look at the following example.

A teacher wants to share 35 kg of clay between 8 groups of students. How much clay will each group get?

Figure _unit2.1.3 Figure 3 Expressed as a decimal: 35 ÷ 8

You can see that each group would get 4.375 kg of clay.

Activity _unit2.1.3 Activity 3: Expressing a remainder as a decimal

Work out the answers to the following without using a calculator.

178 ÷ 4
212 ÷ 5
63 ÷ 8
227 ÷ 4

Answer

44.5
42.4
7.875
56.75

1.2 Interpreting answers when dividing

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

After carrying out a division calculation you may not have an answer that is suitable.

For example, if you were at a restaurant and needed to split a bill of £126.49 between four people you would first calculate the division £126.49 ÷ 4 = £31.6225. Clearly you cannot pay this exact amount and so we would round it up to £31.63 to make sure the whole bill is covered.

In other situations, you may need to round an answer down. If you were cutting a length of wood that is 2 m (200 cm) long into smaller pieces of 35 cm you would initially do the calculation 200 ÷ 35. This would give an answer of 5.714…. As you will only actually be able to get 5 pieces of wood that are 35 cm long, you need to round your answer of 5.714 down to simply 5.

Box _unit2.1.1

Note: The three full stops used in the answer above (5.714…) is a character called an ellipsis. In maths it is used to represent recurring decimal numbers so you don’t have to display them all.

Activity _unit2.1.4 Activity 4: Interpreting answers

Calculate the answers to the following. Decide whether the answers need to be adjusted up or down after calculation of the division sum.

Apples are being packed into boxes of 52. There are 1500 apples that need packing. How many boxes are required?
A bag of flour contains 1000 g. Each batch of cakes requires 150 g of flour. How many batches can you make?
A child gets £2.50 pocket money each week. They want to buy a computer game that costs £39.99. How many weeks will they need to save up in order to buy the game?
A length of copper pipe measures 180 cm. How many smaller pieces that each measure 40 cm can be cut from the pipe?

Answer

1500 ÷ 52 = 28.846 which must be adjusted up to 29 boxes.
1000 ÷ 150 = 6.666 which must be adjusted down to 6 batches.
£39.99 ÷ £2.50 = 15.996 which must be adjusted up to 16 weeks.
180 ÷ 40 = 4.5 which must be adjusted down to 4 pieces.

1.3 Dealing with decimals

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

We often deal with decimals in everyday life, for example, calculations involving money.

Try the following (without a calculator) to check your skills. For this activity, you will need to give your answers in full and only round or adjust your answer if required. If you need a reminder about how to carry out any of the calculations, please refer back to Everyday maths 1.

Activity _unit2.1.5 Activity 5: Calculations with decimals

54.865 + 4.965 + 23.519
6.938 − 5.517
25 + 0.258
54 − 0.65
5.632 × 2.4
1.542 × 1.9
42.4 ÷ 4
39.45 ÷ 1.5
0.48 ÷ 0.025
A factory orders 425 gaskets that weigh 2.3 g each. What is the total weight of the gaskets?
A dispenser holds 15.5 litres of water. How many full 0.2 litre cups of water can you get from one dispenser?
Ahmed and Lea take part in the long jump. Their results are as follows:
- Ahmed 5.501 m
  Lea 5.398 m
  Who jumped furthest and by how much?

Answer

83.349
1.421
25.258
53.35
13.5168
2.9298
10.6
26.3
19.2
977.5 g
77.5 (77 full cups)
Ahmed jumped the furthest by 0.103 m

Summary

In this section you have:

recapped how to carry out calculations with whole numbers and decimals
learned to adjust answers where needed
learned how to express a remainder as a decimal.

2 Dealing with large numbers

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

It is important to be able to carry out calculations with numbers of any size. Large numbers can be written in different ways e.g.

Extract _unit2.2.1

1 200 000 (one million, two hundred thousand) or it can be written as 1.2 million.

Here is another example:

Extract _unit2.2.2

4 250 000 000 (four billion, two hundred and fifty million) is 4.25 billion.

It is often easier to deal with very large numbers when they are written as decimals.

Notice how the decimal is placed after the whole millions or billions.

Hint: A billion is a thousand million.

Using a place value grid can help you to read large numbers as it groups the digits for you, making the whole number easier to read.

Notice how the numbers above are written in this place value grid.

Table _unit2.2.1 Table 1
Billion	Million			Thousand
Billions	Hundreds of millions	Tens of millions	Millions	Hundreds of thousands	Tens of thousands	Thousands	Hundreds	Tens	Units
			1	2	0	0	0	0	0
4	2	5	0	0	0	0	0	0	0

Sometimes when dealing with large numbers it is sensible to round them, for example, the Office for National Statistics gives the number of people unemployed in the UK in February 2019 as 1.36 million. The number of people unemployed will not be exactly 1 360 000 but, by rounding the exact value and writing it as 1.36 million, it is easier to understand.

Activity _unit2.2.1 Activity 6: Rounding large numbers

The following table gives the population of countries.

Round each population to the nearest million and write the figure in shortened form, using decimals where needed.

Table _unit2.2.2 Table 2(a)
Country	Population
UK	66 959 016
China	1 420 062 022

Answer

Table _unit2.2.3 Table 2(b)
Country	Population	Population rounded	Shortened form
UK	66 959 016	67 000 000	67 million
China	1 420 062 022	1 420 000 000	1.42 billion

2.1 Calculations with large numbers

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

The best way to get used to these types of calculations is to go straight into an example.

Case study _unit2.2.1 Example: Calculations with large numbers

Calculate the total population of Malta (0.4 million) and Cyprus (1.2 million).

Method 1

Work in shortened form:

1.2 + 0.4 = 1.6 million

Method 2

Write the numbers in full:

1 200 000 + 400 000 = 1 600 000 (1.6 million)

Activity _unit2.2.2 Activity 7: Calculations with large numbers

Calculate the total turnover for Cambria Trading over the first quarter (3 months).

Table _unit2.2.4 Table 3 Cambria Trading turnover
Month	Profit (£ million)
January	1.2
February	0.9
March	0.85
April	1.1
May	1.02
June	0.87
July	1.19
August	0.98
September	1.05
October	1.08
November	1.8
December	1.65

Calculate the turnover of the last quarter.
Calculate the difference in turnover between the first and last quarters.
- a.Which month had the largest turnover?
- b.Which month had the smallest turnover?
- c.What is the difference between the largest and smallest turnovers?

Answer

Profit in 1st quarter 1.2 + 0.9 + 0.85 = £2.95 million
Profit in last quarter 1.08 + 1.8 + 1.65 = £4.53 million
The difference is 4.53 − 2.95 = £1.58 million
- a.November had the largest turnover at £1.8 million.
- b.March had the smallest at £0.85 million.
- c.The difference is 1.8 − 0.85 = £0.95 million.

Summary

In this section you have learned how to:

write large numbers in full and shortened forms
round large numbers
add and subtract large numbers.

3 Rounding

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Why might you want to round numbers? You may wish to estimate the answer to a calculation or to use a guide rather than work out the exact total. Alternatively, you might wish to round an answer to an exact calculation so that it fits a given purpose, for example an answer involving money cannot have more than two digits after the decimal point.

Figure _unit2.3.1 Figure 4 Rounding up and down

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Illustration of an up arrow and a down arrow

Figure 4 Rounding up and down

You will now explore each of these examples in more detail and practise your rounding skills in context.

3.1 Rounding to a degree of accuracy

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Watch the short video below to see an example of how to round to 1, 2 and 3 decimal places.

Download this video clip.Video player: bltl_1_2_3_rounding.mp4

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Transcript

Rounding can be used when you're asked to give an answer to a given degree of accuracy. This is called rounding to decimal places, or d.p. In this video, you'll look at how to round to one, two, and three decimal places.

Take the number 25.782. How would you write this rounded to one decimal place? This is the first decimal place. Rounded to 1 d.p., the number could be either 25.7 or 25.8. Look at the digit next to the number 7. If this number is equal to or greater than 5, add one more. If it's less than 5, leave it. In this case, 8 is greater than 5, so our number rounded to 1 d.p. is 25.8.

Now, let's around a number to two decimal places. A common example of when you would do this in daily life is when dealing with money. If you were at a restaurant and needed to split a bill of £87.95 between three people, you would first calculate the division. £87.95 divided by 3 equals £29.3166667. Clearly, you cannot pay this exact amount. So how much would you pay if the amount per person was rounded to two decimal places?

This is the second decimal place. Look at the number next to it. Is it greater than or equal to 5? 6 is more than 5, so you need to add 1. The amount to pay is £29, 32 pence.

Let's try another example. How would you around the number 35.496 to two decimal places? This is the second decimal place. Look at the number next to it. 6 is greater than 5, so you need to add one more. In this case, the rounded number would be 35.50.

Can you round to three decimal places? Have a go with this number: 412.5762. The number next to the third decimal place is 2, which is less than 5. This means the correctly rounded number is 412.576.

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Remember this rounding rhyme to help you:

Figure _unit2.3.2 Figure 5 A rounding rhyme

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A rounding rhyme. The text reads: Underline the digit, look next door. If it’s 5 or greater, add one more. If it’s less than 5, leave it for sure. Everything after is a zero, not more.

Figure 5 A rounding rhyme

Activity _unit2.3.1 Activity 8: Rounding skills

Practise your rounding skills by completing the below.

What is 24.638 rounded to one decimal place?
What is 13.4752 rounded to two decimal places?
What is 203.5832 rounded to two decimal places?
What is 345.6794 rounded to three decimal places?
What is 3.65 rounded to the nearest whole number?
What is £199.755 to the nearest penny?
What is £37.865 to the nearest pound?
What is 61.607 kg to the nearest kg?

Answer

24.6
13.48
203.58
345.679
4
£199.76
£38
62 kg

3.2 Rounding to approximate an answer

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

You might round in order to approximate an answer. At the coffee shop, you might want to buy a latte for £2.85, a cappuccino for £1.99 and a tea for £0.99. It is natural to round these amounts up to £3, £2 and £1 in order to arrive at an approximate cost of £6 for all three drinks. It is also very useful when checking calculations to make sure that your answer makes sense, especially when working with large numbers and decimals.

Activity _unit2.3.2 Activity 9: Approximation

Calculate the following using a calculator and use estimation to check your answers.

On 5th March 2019, 190 people matched 5 numbers and won £1650 each. What was the total prize fund?
Swansea AFC’s home ground is the Liberty Stadium which holds 21 088 fans. Cardiff City FC plays at the Cardiff City Stadium which holds 33 316 fans.
What is the difference in capacity at these grounds?

Answer

The actual answer is £313 500 (190 × 1650)
Estimate 200 × 1700.
2 × 17 = 34 so 200 × 1700 = 340 000 so the answer is sensible.
The actual answer is 12 228 (33 316 − 21 088)
Estimate 33 000 − 21 000 = 12 000 so the answer is sensible.

Summary

In this section you have learned:

how and when to use rounding to approximate an answer to a calculation
how to round an answer to a given degree of accuracy – e.g. rounding to two decimal places.

4 Multistage calculations

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Often in daily life you will come across problems that require more than one calculation to reach the final answer.

Case study _unit2.4.1 Example: Multistage calculations

Four friends are planning a holiday. The table below shows the costs:

Table _unit2.4.1 Table 4
Item	Price
Flight (return)	£305 per person
Taxes	£60 per person
Hotel	£500 per room. 2 people per room
Taxi to airport	£45

The friends will be sharing the total cost equally between them. How much do they each pay?

Method

First we use multiplication to find the cost of items that we need more than one of:

Flights = £305 × 4 = £1220
Taxes = £60 × 4 = £240
Hotel = 2 rooms required for 4 people = £500 × 2 = £1000

Now we use addition to add these totals together along with the taxi fare:

£1220 + £240 + £1000 + £45 = £2505

Finally, we need to use division to find out how much each person pays:

£2505 ÷ 4 =£626.25 each

Activity _unit2.4.1 Activity 10: Multistage calculations

Your current mobile phone contract costs you £24.50 per month.
You are considering changing to a new provider. This provider charges £19.80 per month along with an additional, one off connection fee of £30.
How much will you save over the year by switching to the new provider?
You are going on holiday and you have decided to stay in a cottage in North Wales for 7 nights.
There will be 12 of you staying and the total cost of the cottage you have chosen is £460 per night. If you split the cost equally, how much will each of you pay?

Answer

First calculate the cost with the current provider
monthly cost × 12
£24.50 × 12 = £294 (current provider)
Second, calculate the cost of the new provider.
To do this you need to calculate the total monthly costs:
- £19.80 × 12 = £237.60
and then add on the one off connection fee of £30:
- £237.60 + £30 = £267.60 (new provider)
Finally you can calculate the difference between the two providers:
- £294 − £267.60 = £26.40 saved
To do your calculation, you need to work out the cost for 7 nights. Once you have done this, you can divide the total by 12:
- 460 × 7 = £3220
  £3220 ÷ 12 = £268.33 (rounded to two d.p.)
  Here we would probably round this amount up to £268.34 per person to make sure the full cost is covered.

All the examples we have looked at until now have used positive numbers. However, as anyone with an overdraft will know, numbers (or bank balances) are not always positive! Our next section therefore deals with negative numbers.

Summary

In this section you have:

applied the four operations to solve multistage calculations.

5 Negative numbers

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Negative numbers come into play in two main areas of life: money and temperature. Watch the animations below for some examples.

Download this video clip.Video player: bltl_1_3_negative.mp4

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Transcript

Negative numbers come into play in two main areas of life: money and temperature. If your bank balance is showing as -£250, this means that your balance is £250 below zero. In other words, you owe the bank £250.

Similarly, you'll be used to seeing winter weather forecasts predicting temperatures of -10 degrees Celsius. This means the temperature will be 10 degrees below zero degrees Celsius.

It can be useful to think of a thermometer if you're trying to find the difference between two numbers, where one or both of the numbers are negative. If the temperature is increasing, you go up the thermometer. If the temperature is decreasing, you travel down the thermometer.

For example, you'd like to know the answer to the sum [6 - 10]. Starting at number 6 on a thermometer and travelling down, it's easy to see that subtracting 10 will result in a number less than zero.

What is the answer to the sum [-5 + 12]? This time, we're starting below zero and travelling up the thermometer by 12 places, giving an answer of 7.

Can you work out the answer to the sum [-1 - 8]? In this example, we start below zero and the answer is also below zero: -9. Now try the next activity for more practise with negative numbers.

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Activity _unit2.5.1 Activity 11: Negative and positive temperature

The table below shows the temperatures of cities around the world on a given day

Table _unit2.5.1 Table 5
London	Oslo	New York	Kraków	Delhi
4˚C	−12C	7˚C	−3˚C	19˚C

- a.Which city was the warmest?
- b.Which city was the coldest?
- c.What is the difference in temperature between the warmest and coldest cities?

Answer

- a.Delhi was the warmest city as it has the highest positive temperature.
- b.Oslo was the coldest city as it has the largest negative temperature.
- c.The difference between the temperatures in these cities is 31˚C.
  From 19˚C down to 0˚C is 19˚C and then you need to go down a further 12˚C to get to −12˚C.

Look at this bank statement.
Figure _unit2.5.1 Figure 6 A bank statement
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A bank statement with four columns: Date, Description, Amount and Balance. There are six lines under the column headings showing money paid in and out. (Date column): 09 Oct, 11 Oct, 15 Oct, 20 Oct, 21 Oct, 25 Oct. (Description column): Bank Transfer, Direct Debit, Automated Pay In, Bank Bank Transfer, Direct Debit, Bank Transfer. The (Amount column) is blank. (Balance column): 100.00, −20.00, 70.00, 100.00, −50.00, 200.00.
Figure 6 A bank statement
- a.On which days was Sonia Cedar overdrawn, and by how much?
- b.How much money was withdrawn between 9 and 11 of October?
- c.How much was added to the account on 15 October?

Answer

- a.The minus sign (−) indicates that the customer is overdrawn, i.e. owes money to the bank.
  The amount shows how much they owe. So Sonia Cedar was overdrawn on 11 October by £20 and by £50 on 21 October.
- b.£120 was withdrawn on 11 October.
  The customer had £100 in the account and must have withdrawn another £20 (i.e. £100 + £20 = £120 in total) in order to be £20 overdrawn.
- c.The customer owed £20 and is now £70 in credit, so £90 must have been added to the account.

Look at the table below showing a company’s profits over 6 months.
Hint: a negative profit means that the company made a loss.

Table _unit2.5.2 Table 6
Month	Profit (£000)
January	166
February	182
March	−80
April	124
May	98
June	−46
Balance

- a.Which month had the greatest profit?
- b.Which month has the greatest loss?
- c.What was the overall balance for the six months?
  Hint: start by calculating the total profits and the total losses.

Answer

- a.February had the largest profit with £182 000 (remember to look at the column heading which shows that the figures are in 000s – thousands).
- b.March showed the greatest loss at £80 000.
- c.To calculate the overall balance you need to first calculate the total profits and the total losses. To calculate the profits you need to do this calculation:
- 166 + 182 + 124 + 98 = 570
So the profit was £570 000.
Next you need to calculate the total losses; two months showed a loss so you need to add these values:
- 80 + 46 = 126 so the losses over the six months were £126 000.
Now you can calculate the overall balance by subtracting the losses from the profits:
- £570 000 − £126 000 = £444 000
This is a positive value so it means the company made an overall profit of £444 000.

Summary

In this section you have:

learned the two main contexts in which negative numbers arise in everyday life – money (or debt!) and temperature
practised working with negative numbers in these contexts.

6 Mathematical terms

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

It is important to know the meaning of the following terms:

multiples
lowest common multiple
factors
common factors
prime numbers

Multiples

A multiple of a number can be found by multiplying that number by any whole number e.g. multiples of 2 include 2, 4, 6, 8, 10 etc. (all are in the 2 times table).

Box _unit2.6.1

Note: To check if a number is a multiple of another, see if it divides exactly into the multiple, e.g. to see if 3 is a multiple of 81 do 81 ÷ 3 = 27. It divides exactly so 81 isa multiple of 3.

Lowest common multiple

In maths, we sometimes need to find the lowest common multiple of numbers.

The lowest common multiple (LCM) is simply the smallest multiple that is common to more than one number.

Case study _unit2.6.1 Example: Lowest common multiple of 3 and 5

Hint: when looking for multiples, it is easiest to start by listing the multiples of the highest number first. This saves you going any further than you need to with the list.

The first few multiples of 5 are:

5, 10, 15, 20, 25, 30 etc.

The first few multiples of 3 are:

3, 6, 9, 12, 15, 18, 21 etc.

You can see that the lowest number that is a common multiple of 3 and 5 is 15.

Activity _unit2.6.1 Activity 12: Finding the lowest common multiple

Find the lowest common multiple of:

6 and 12
2 and 7

Answer

The lowest common multiple of 6 and 12 is 12:
Multiples of 12:
12, 24, 36, 48, 60 etc.
Multiples of 6:
6, 12, 18, 24, 30 etc.
You can see from the list that 24 is also a common multiple of 6 and 12, but 12 is the lowest common multiple.
The lowest common multiple of 2 and 7 is 14:
Multiples of 7:
7, 14, 21, 28, 35, 42 etc.
Multiples of 2:
2, 4, 6, 8, 10, 12, 14 etc.

Factors, common factors and prime numbers

Factors of a number divide into it exactly. Factors of all numbers include 1 and the number itself. However, most numbers have other factors as well. If you think of all of the numbers that multiply together to make that number, you will find all of the factors of that number.

Case study _unit2.6.2 Example: What are the factors of 8?

8 × 1 = 8

2 × 4 = 8

So the factors of 8 are 1, 2, 4 and 8.

Activity _unit2.6.2 Activity 13: Finding factors

What are the factors of 54?
What are the factors of 165?

Answer

The factors of 54 are 1, 2, 3, 6, 9, 18, 27 and 54
The factors of 165 are 1, 3, 5, 11, 15, 33, 55 and 165

A common factor is a factor that goes into more than one number. For example, 4 is a common factor of 8 and 12 because it divides exactly into both numbers.

Prime Numbers

A prime number is a number which only has 2 factors: 1 and itself.

The prime numbers between 1 and 20 are 2, 3, 5, 7, 11, 13, 17 and 19.

Box _unit2.6.2

Note:

1 is not a prime number as it only has one factor.
2 is the only even prime number.

You have now learned how to use all four operations, how to work with negative numbers and learned some important mathematical terms. Every other mathematical concept hinges around what you have learned so far; so once you are confident with these, you’ll be a success!

Summary

In this section you have:

learned some key mathematical terms: multiple, lowest common multiple, factor, common factor and prime number
identified the lowest common multiple
identified factors.

7 Fractions

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

You will be used to seeing fractions in your everyday life, particularly when you are out shopping or scouring the internet for the best deals. It’s really useful to be able to work out how much you’ll pay if an item is on sale or if a supermarket deal really is a good deal!

Figure _unit2.7.1 Figure 7 A poster advertising a sale

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An illustration of a sale poster reading ‘Sale, many lines 1/3 off original price’.

Figure 7 A poster advertising a sale

There are several different elements to working with fractions. First you will look at simplifying fractions.

7.1 Simplifying fractions

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Watch the video below which looks at how to simplify fractions before having a go yourself in Activity 14.

Download this video clip.Video player: bltl_1_4_2_simplifying_fractions.mp4

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Transcript

In this video, you'll look at how to simplify fractions. You'll be used to seeing the results of company surveys given as fractions: '7/8 people say they were satisfied with our customer service'. This is an example of a fraction in its simplest form. It could be that 184 people were surveyed and that 161 of them responded to say they were satisfied. Although they are equivalent fractions, you can see that the fraction '7/8 people' is a lot more user friendly than saying 161/184 people responded to say they were satisfied.

If you're asked to give an answer as a fraction in its simplest form, you firstly find a number that you can divide both parts of the fraction by, and divide it. For example, to simplify the fraction 12/18, you can divide both the top and the bottom number by 2, to give 6/9.

Keep going until you can't find a number that you can divide both parts of the fraction by. The fraction 6/9 can be divided again, this time by 3, to get 2/3. This is now the fraction in its simplest form.

Now let's simplify the fraction 40/120. Remember, find a number that you can divide both parts of the fraction by, and keep going until there is no longer a number that works. So, divide the top and bottom by 10. Divide the top and bottom by 2. Divide the top and bottom by 2. This is the fraction in its simplest form.

End transcript

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Activity _unit2.7.1 Activity 14: Simplifying fractions

Show the following fractions in simplest form, where possible:

Answer

a. =
b. =
c. =
d. =
e. can’t be simplified
f. =

Next you’ll look at expressing a quantity of an amount as a fraction.

7.2 Writing a quantity of an amount as a fraction

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Sometimes you will need to show one amount as a fraction of another. This might sound complicated, but it’s actually very logical. Look at the examples below.

Case study _unit2.7.1 Example 1: Fraction of an amount

In Figure 8, what fraction of Smarties are red?

Figure _unit2.7.2 Figure 8 Smarties in different colours

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A quantity of Smarties in different colours. 4 are red; 6 are blue; 5 are pink; 3 are green; 6 are yellow; 2 are orange; 4 are purple.

Figure 8 Smarties in different colours

Extract _unit2.7.1

To express the fraction of Smarties that are red, you simply need to count the red Smarties (4) and the total number of Smarties (30). Since there are 4 red Smarties out of 30 altogether, the fraction is . It is worth noting here that this could also be written as 4/30.

You may well be asked to give your answer as a fraction in its simplest form, so always check to see if you can simplify your answer. In this case will simplify to .

Case study _unit2.7.2 Example 2: Fraction of an amount

250 g of flour is taken from a 1 kg bag. What fraction is this?

Hint: there are 1000 g in a kg.

To express quantities as fractions, the top and bottom numbers need to be in the same units, so here you need to make sure that you express both the top and bottom values in grams:

The flour removed is already expressed in grams: 250 g
The total amount is in kilograms so you need to convert to grams: 1 kg = 1000 g

Now write the amount taken over the total amount to express as a fraction:

(250 g of flour out of the 1000 g bag has been taken or used)

Then cancel down (or simplify) if possible:

So of the flour has been used.

Activity _unit2.7.2 Activity 15: Expressing one number as a fraction of another

What fraction of a kilogram is:
- a.100 g
- b.750 g
- c.640 g
- d.20 g

Answer

100 g = 1000 g, so:
- a.100 g = of a kilogram
- b.750 g = of a kilogram
- c.640 g = of a kilogram
- d.20 g = = of a kilogram

What fraction of an hour is:
- a.15 minutes
- b.20 minutes
- c.35 minutes
- d.48 minutes

Answer

1 hour = 60 minutes so:
- a.15 minutes = = of an hour.
- b.20 minutes = = of an hour.
- c.35 minutes = = of an hour.
- d.48 minutes = = of an hour.

A farmer takes 120 eggs to the local farmer’s market. She has 24 eggs left at the end of the day. What fraction of the eggs are left?

Answer

are left. This cancels down to , so of the eggs are left.

A class of students sit a test. 18 pass and 12 fail. What fraction passed the test?

Answer

Work out the total number of students by adding the number who passed to those who failed 18 + 12 = 30. Now work out the fraction that passed:
- (18 out of 30 students passed)
Now cancel down:
- = .
  So passed the test.

Mary bought her car for £12 500. When she goes to trade it in she is offered £8750. What fraction of the original price is this?

Answer

30 people entered a raffle. 6 of these people won a prize. What fraction of people did not win a prize? Give your answer as a fraction in its simplest form.

Answer

As this question wants the number of people who did not win a prize we must first do:
30 − 6 = 24 people did not win a prize.
As a fraction this becomes which simplifies to .

Sometimes fractions will not cancel down easily. When this happens, you estimate the fraction by rounding the numbers to values that will cancel. Sometimes this means breaking the ‘rules’ of rounding.

Case study _unit2.7.3 Example: Estimating fractions

will not cancel.

By rounding 1347 up to 1400 and 2057 to 2100 we can cancel down the fraction to get:

Box _unit2.7.1

Note: Round to numbers that are easy to cancel, but if you round off too much, you will lose the accuracy of your answer.

Now that you can express a quantity as a fraction, estimate and simplify fractions, the next step is to be able to work out fractions of amounts. For example, if you see a jacket that was priced at £80 originally but is in the sale with off, it’s useful to be able to work out how much you will be paying.

7.3 Fractions of amounts

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Fractions of amounts can be found by using your division and multiplication skills. To work out a fraction of any amount you first divide your amount by the number on the bottom of the fraction – the denominator. This gives you 1 part.

You then multiply that answer by the number on the top of the fraction – the numerator.

It is worth noting here that if the number on the top of the fraction is 1, multiplying the answer will not change it so there is no need for this step. Take a look at the examples below.

Case study _unit2.7.4 Example: Divide by the denominator

Method

To find of 90 we do 90 ÷ 5 = 18.

Since the number on the top of our fraction is 1, we do not need to multiply 18 by 1 as it will not change the answer.

So of 90 = 18.

Case study _unit2.7.5 Example: Multiply by the numerator

Method

To find of 42 we do 42 ÷ 7 = 6.

This means that of 42 = 6.

Since you want of 42, we then do 6 × 4 = 24.

So of 42 = 24.

Let’s go back to the jacket that used to cost £80 but is now in the sale with off. How do you find out how much it costs? Firstly, you need to find of 80. To calculate this you do:

£80 ÷ 5 = £16 and then £16 × 2 = £32

This means that you save £32 on the price of the jacket. To find out how much you pay you then need to do £80 − £32 = £48.

You will have practised finding fractions of amounts in Everyday maths 1, but have a go at the following activity to recap this important skill.

Activity _unit2.7.3 Activity 16: Finding fractions of amounts

Work out the following without using a calculator. You may double-check on a calculator if you need to and remember to check your answers against ours.

You are looking to buy house insurance and want to get the best deal. Put the following offers in order, from cheapest to most expensive, after the discount has been applied.

Table _unit2.7.1 Table 7
Company A	Company B	Company C
£120 per year	£147 per year	£104 per year
Special offer: off!	Special offer: off!	Special offer: off!

Answer

Company C is cheapest:
- of £104 = £104 ÷ 4 = £26 discount
- £104 − £26 = £78
Company A is second cheapest:
- of £120 = £120 ÷ 3 = £40 discount
  £120 − £40 = £80
Company B is most expensive:
- of £147 = £147 ÷ 7 × 2 = £42 discount
- £147 − £42 = £105

A cinema sells 2400 tickets over a weekend. They review their ticket sales and find that of the weekend ticket sales were to adults. How many adult tickets were sold?

Answer

1600 tickets sold to adults:
- 2400 ÷ 3 = 800 to give
- 2 × 800 = 1600 to give

A college has raised of its £40 000 charity fundraising target. How much money does the college need to raise to meet its target?

Answer

£16 000 needed to meet target.
40 000 ÷ 5 = 8000 to give
8000 × 3 = 24 000 to give (the amount raised)
But the question asks how much is needed to meet its target so we need to subtract the amount raised from the target:
- 40 000 − 24 000 = £16 000

Discounts and special offers are not always advertised using fractions. Sometimes, you will see adverts with 10% off or 15% off. Another common area where we see percentages in everyday life would be when companies apply VAT at 20% to items or when a restaurant adds a 12.5% service charge. The next section looks at what percentages are, and how to calculate them.

Summary

In this section you have:

learned how to express a quantity of an amount in the form of a fraction
learned how to, and practised, simplifying fractions
revised your knowledge on finding fractions of amounts.

8 Percentages

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

There are different ways of working out percentages of amounts. We will take a look at the most common methods now.

Figure _unit2.8.1 Figure 9 Percentage discounts in a sale

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A series of tags sequentially reading ‘Sale up to 10%, 15%, 20%, 25%, 30%, 40%, 50%’.

Figure 9 Percentage discounts in a sale

Box _unit2.8.1

Note: You may use a different method to these ones. You may even use different methods depending on the percentage you are calculating. Do whatever works for you.

8.1 Calculating a percentage of an amount

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Method 1

Percentages are just fractions where the number on the bottom of the fraction must be 100. If you wanted to find out 15% of 80 for example, you work out of 80, which you already know how to do!

Working out the percentage of an amount requires a similar method to finding the fraction of an amount. Take a look at the examples below to increase your confidence.

Case study _unit2.8.1 Example 1: Finding 17% of 80

Method

17% of 80 = of 80, so here we do:

Extract _unit2.8.1

80 ÷ 100 = 0.8

0.8 × 17 = 13.6

Another way of thinking about this method is that you are dividing by 100 to find 1% first and then you are multiplying by whatever percentage you want to find.

Alternatively, you could multiply the value by the top number first and then divide by 100:

Extract _unit2.8.2

17 × 80 = 1360

1360 ÷ 100 = 13.6

The answer will be the same.

Case study _unit2.8.2 Example 2: Finding 3% of £52.24

Method

3% of 52.24 = of 52.24, so we do:

Extract _unit2.8.3

52.24 ÷ 100 = 0.5224.

0.5224 × 3 = £1.5672 (£ 1.57 to two d.p.)

Extract _unit2.8.4

52.24 × 3 = 156.72

156.72 ÷ 100 = £1.5672 (£1.57 to two d.p.)

This is a good method if you want to be able to work out every percentage in the same way. It can be used with and without a calculator. Many calculators have a percentage key, but different calculators work in different ways so you need to familiarise yourself with how to use the % button on your calculator.

Method 2

To use this method you only need to be able to work out 10% and 1% of an amount. You can then work out any other percentage from these.

Let’s just recap how to find 10% and 1%.

Box _unit2.8.2 10%

To find 10% of an amount divide by 10:

Extract _unit2.8.5

10% of £765 = 765 ÷ 10 = £76.50

10% of £34.50 = 34.50 ÷ 10 = £3.45

Hint: remember to move the decimal point one place to the left to divide by 10.

Box _unit2.8.3 1%

To find 1% of an amount divide by 100:

Extract _unit2.8.6

1% of £765 = 765 ÷ 100 = £7.65

1% of £34.50 = 34.50 ÷ 100 = £0.345 (£0.35 to two d.p.)

Hint: remember to move the decimal point two places to the left to divide by 100.

Once you know how to work out 10% and 1%, you can work out any other percentage.

Case study _unit2.8.3 Example 1: Finding 24% of 60

Find 10% first:

Extract _unit2.8.7

60 ÷ 10 = 6

10% = 6

20% is 2 lots of 10% so:

Extract _unit2.8.8

6 × 2 = 12

20% = 12

Now find 1%:

Extract _unit2.8.9

60 ÷ 100 = 0.6

4% is 4 lots of 1% so:

Extract _unit2.8.10

0.6 × 4 = 2.4

4% = 2.4

Now add the 20% and 4% together:

12 + 2.4 = 14.4

Case study _unit2.8.4 Example 2: Finding 17.5% of £328

17.5% can be broken up into 10% + 5% + 2.5%, so you need to work out each of these percentages and then add them together.

Find 10% first:

Extract _unit2.8.11

328 ÷ 10 = 32.8

10% = 32.8

5% is half of the 10% so:

Extract _unit2.8.12

32.8 ÷ 2 = 16.4

5% = 16.4

2.5% is half of the 5% so:

Extract _unit2.8.13

16.4 ÷ 2 = 8.2

2.5% = 8.2

Now add the 10%, 5% and 2.5% figures together:

Extract _unit2.8.14

32.8 + 16.4 + 8.2 = £57.40

This is a good method to do in stages when you do not have a calculator.

Box _unit2.8.4

Note: There are some other quick ways of working out certain percentages:

Extract _unit2.8.15

50% – divide the amount by two.

25% – halve and halve again.

These quick facts can be used in combination with method 2 to make calculations, e.g. 60% could be worked out by finding 50%, 10% and then adding the 2 figures together. You just need to look for the easiest way to split up the percentage to make your calculation.

Activity _unit2.8.1 Activity 17: Finding percentages of amounts

Use whichever method/s you prefer to calculate the answers to the following:

Find:
- a.45% of £125
- b.15% of 455 m
- c.52% of £677
- d.16% of £24.50
- e.2% of 4000 kg
- f.82% of £7.25
- g.37% of £95
The Cambria Bank pays interest at 3.5%. What is the interest on £3000?
Sure Insurance offer a 30% No Claims Bonus. How much would be saved on a premium of £345.50?
Sunshine Travel Agents charge 1.5% commission on foreign exchanges. What is the charge for changing £871?

Answer

- a.£56.25
- b.68.25 m
- c.£352.04
- d.£3.92
- e.100 kg
- f.£5.945 (£5.95 to 2 d.p.)
- g.£35.625 (£35.63 to two d.p.)
£105
£103.65
£13.065 (£13.07 to two d.p.)

Just as with fractions you will often need to be able to work out the price of an item after it has been increased or decreased by a given percentage. The process for this is the same as with fractions; you simply work out the percentage of the amount and then add it to, or subtract it from, the original amount.

Activity _unit2.8.2 Activity 18: Percentages increase and decrease

You earn £500 per month. You get a 5% pay rise.
- a.How much does your pay increase by?
- b.How much do you now earn per month?

Answer

- a.£25
- b.£525 per month.

You buy a new car for £9500. By the end of the year its value has decreased by 20%.
- a.How much has the value of the car decreased by?
- b.How much is the car worth now?

Answer

- a.The car has decreased by £1900.
- b.The car is now worth £7600.

You invest £800 in a building society account which offers fixed-rate interest at 4% per year.
- a.How much interest do you earn in one year?
- b.How much do you have in your account at the end of the year?

Answer

- a.£32 interest earned.
- b.£832 in the account at the end of the year.

Last year Julie’s car insurance was £356 per annum. This year she will pay 12% less. How much will she pay this year?

Answer

She will pay £42.72 less so her insurance will cost £313.28

A zoo membership is advertised for £135 per year. If Tracy pays for the membership in full rather than in monthly installments, she receives a 6% discount. How much will she pay if she pays in full?

Answer

She will save £8.10 so she will pay £126.90.

A museum had approximately 5.87 million visitors last year. Visitor numbers are expected to increase by 4% this year. How many visitors is the museum expecting this year?

Answer

5.87 million = 5 870 000.
4% of 5 870 000 = 234 800
5 870 000 + 234 800 = 6 104 800 people

Next you’ll look at how to express one number as a percentage of another.

8.2 Expressing one number as a percentage of another

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Sometimes you need to write one number as a percentage of another. You have already practised writing one number as a fraction of another; this just takes it a bit further.

Case study _unit2.8.5 Example 1: What percentage are women?

A class is made up of 21 women and 14 men, what percentage of the class are women?

To work this out, you start by expressing the numbers as a fraction. You then multiply by 100 to express as a percentage.

The formula is:

× 100

In this case, 21 out of a total of 35 people are women so the sum we do would be:

× 100

The fraction line is also a divide line, so if you were doing this on a calculator you would do:

21 ÷ 35 × 100 = 60%

How would you work this out without a calculator?

There are different ways you can make the calculation. Two methods are shown below.

Method 1

You start by multiplying the top number in the fraction by 100. The bottom number will stay the same:

Now you need to cancel the fraction down as much as possible:

÷ top and bottom by 5 = , then, ÷ top and bottom by 7 =

Anything over 1 is a whole number so the answer is 60.

So 60% of the class are women.

Box _unit2.8.5

Note: When using this method, if you cancel as far as possible and you do not end up with an answer over 1, you will need to divide the top number by the bottom number to work out the final answer, e.g. the fraction cannot cancel any further, so:

15 ÷ 4 = 3.75

Method 2

The other method involves expressing the fraction as a decimal first and then converting it to a percentage. This means that you multiply by 100 at the very end of the calculation.

A local attraction sold 150 tickets last bank holiday, 102 of which were full price. What percentage of the tickets sold were at the concessionary price?

Work out the number of concessionary tickets sold:

150 – 102 = 48

Write the number of concessionary tickets sold as a fraction of the total number sold:

Cancel down your fraction:

÷ top and bottom by 6 =

Once you cannot cancel any further, you need to divide the top number by the bottom number to express as a decimal:

8 ÷ 25 = 0.32

Figure _unit2.8.2 Figure 10 Expressed as a decimal: 8 divided by 25

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8.00 ÷ 25 = 0.32. This illustration shows that the sum has been handrawn and there is a small ‘8’ above the first ‘0’ of ‘8.00’ and a small ‘5’ above the second ‘0’of 8.00.

Figure 10 Expressed as a decimal: 8 divided by 25

Finally, multiply the decimal answer by 100 to express as a percentage:

0.32 × 100 = 32%

So 32% of the tickets were sold at the concessionary price.

Activity _unit2.8.3 Activity 19: Expressing one number as a percentage of another

Use whichever method you prefer to calculate the answers to the following. Give answers to two d.p. where appropriate.

Hint: make sure your units are the same first.

What percentage:
- a.of 1 kg is 200 g?
- b.of an hour is 48 minutes?
- c.of £6 is 30p?

Answer

- a.1 kg = 1000 g
  × 100 = 20%
- b.1 hour = 60 minutes
  × 100 = 80%
- c.£1 = 100 p
  × 100 = 5%

Bea swam 50 laps of a 25 m swimming pool in a charity swim. A mile is almost 1600 m. What percentage of a mile did Bea swim?

Answer

50 × 25 = 1250 m
× 100 = 78.13% (to two d.p.)

A student gets the following results in the end of year tests:

Table _unit2.8.1 Table 8
	Maths	English	Science	Art
Mark achieved	64	14	72	56
Possible total mark	80	20	120	70

Calculate her percentage mark for each subject.

Answer

Maths: × 100 = 80%

English: × 100 = 70%

Science: × 100 = 60%

Art: × 100 = 80%

Susan is planting her flower beds. She plants 13 yellow flowers, 18 white flowers and 9 red ones. What percentage of her flowers will not be white?

Answer

Number not white = 13 + 9 = 22
Total number she is planting = 13 + 18 + 9 = 40
× 100 = 55%
55% of the flowers will not be white.

A building society charges £84 interest on a loan of £1200 over a year. What percentage interest is this?

Answer

× 100 = 7%
The interest rate is 7%.

Next you will look at percentage change. This can be useful for working out the percentage profit (or loss) or finding out by what percentage an item has increased or decreased in value.

8.3 Percentage change

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Watch the video below on how to calculate percentage change, then complete Activity 20.

Download this video clip.Video player: bltl_1_5_2_percentage.mp4

Show transcript|Hide transcript

Transcript

You will already be familiar with the concept that if you buy a new car, when you come to sell it, its value is likely to have decreased. On the other hand, if you're lucky, when you buy a house and then wish to sell, you may be able to sell it for more than you bought it for.

The percentage difference between the original price and the sale price is called percentage change. This can be useful for working out the percentage profit, or loss, or finding out by what percentage an item has changed in value. It's important to recognise that percentage change can be either positive, if the price has increased, or negative, if the value of the item has gone down.

To calculate the percentage change, we need to use the simple formula: percentage change equals the difference, divided by the original, multiplied by 100. 'Difference' refers to the difference between the two values. That is, the cost before and after. 'Original' means the original value of the item.

Let's use this formula in an example. A train ticket used to cost £24. The price has gone up to £27.60. How do we find the percentage increase? First, you need to find the difference in cost. The difference = £27.60 - £24, which is £3.60.

Next, divide this difference by the original cost, and then multiply by 100. £3.60 divided by £24, x 100, = 15. The cost of the train ticket has increased by 15%.

Now try another example. You bought a car for £4,500 and you managed to sell it for £3,200. What is the percentage decrease? You should give your answer to two decimal places. The difference is £4,500 - £3,200, which = £1,300. Divide this by the original cost of £4,500. 1,300 divided by 4,500, x 100, equals a 28.89% per cent decrease. Now practise using the percentage change formula in the next activity.

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Activity _unit2.8.4 Activity 20: Percentage change formula

Practise using the percentage change formula which you learned about in the video above on the four questions below. Where rounding is required, give your answer to two decimal places.

Last year your season ticket for the train cost £1300. This year the cost has risen to £1450. What is the percentage increase?

Answer

Difference: £1450 − £1300 = £150
- Original: £1300
- Percentage change = × 100
- Percentage change = 0.11538... × 100 = 11.54% increase (rounded to two d.p.)

You bought your house 10 years ago for £155 000. You are able to sell your house for £180 000. What is the percentage increase the house has made?

Answer

Difference: £180 000 − £155 000 = £25 000
- Original: £155 000
- Percentage change = × 100
- Percentage change = 0.16129... × 100 = 16.13% increase (rounded to two d.p.)

You purchased your car 3 years ago for £4200. You sell it to a buyer for £3600. What is the percentage decrease of the car?

Answer

Difference: £4200 − £3600 = £600
- Original: £4200
- Percentage change = × 100
- Percentage change = 0.14285... × 100 = 14.29% decrease (rounded to two d.p.)

Stuart buys a new car for £24 650. He sells it 1 year later for £20 000. What is the percentage loss?

Answer

Difference: £24 650 − £20 000 = £4650
Original: £24 650
Percentage change = × 100
4650 ÷ 24 650 × 100 = 18.86% loss (rounded to two d.p.)

Congratulations, you now know everything you need to know about percentages! As you have seen, percentages come up frequently in many different areas of life and having completed this section, you now have the skills to deal with all kinds of situations that involve them.

You saw at the beginning of the section that percentages are really just fractions. Decimals are also closely linked to both fractions and percentages. In the next section you will see just how closely related these three concepts are and also learn how to convert between each of them.

Summary

In this section you have:

found percentages of amounts
calculated percentage increase and decrease
calculated percentage change using a formula
expressed one number as a percentage of another.

9 Fractions, decimals and percentages

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

You have already worked with decimals in this course and many times throughout your life. Every time you calculate something to do with money, you are using decimal numbers. You have also learned how to round a number to a given number of decimal places.

Figure _unit2.9.1 Figure 11 Equivalent decimals, fractions and percentages

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Illustration showing equivalences: Decimal, 0.5; Fraction, 1/2; Percentage, 50%.

Figure 11 Equivalent decimals, fractions and percentages

9.1 Converting between percentages, decimals and fractions

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Since fractions, decimals and percentages are all just different ways of representing the same thing, we can convert between them in order to compare. Take a look at the video below to see how to convert fractions, decimals and percentages.

Download this video clip.Video player: bltl_1_6_fractions_decimals_percentages.mp4

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Transcript

Equivalent fractions, decimals, and percentages are all just different ways of representing the same thing, so you can convert between them to compare. First, let's look at turning percentages into decimals. Take the example of 60%. Remember that a percentage is out of 100. To turn a percentage into its equivalent decimal, you need to divide by 100. 60 divided by 100 = 0.6. So 0.6 is the equivalent decimal of 60%.

What about 25%? Can you work out the equivalent decimal of 25% before the answer is revealed? Remember that 'cent' means 100.

To turn 25% into its equivalent decimal, you need to divide 25 by 100, which = 0.25.

Now let's find the equivalent fraction for the decimals we've calculated. To do this, you need to first turn it into a percentage and write it as a fraction out of 100. Using our previous examples, 0.6 or 60%, can be written as 60 over 100. 0.25, or 25%, can be written as 25 out of 100.

However, these fractions are not written in their simplest form. Remember that to simplify fractions, you need to divide both parts by the same number and keep going until you can't find a number that you can divide both parts by. In their simplest form, 0.6 is written as the fraction 3/5. 0.25 is written as the fraction one quarter, or 1 over 4.

Now looking at the table, you can see how percentages, decimals and fractions relate to each other.

Let's try one more example. Can you work out the equivalent decimal and fraction of 72%? 72 divided by 100 gives the decimal 0.72. To find the equivalent fraction, you need to change it into a percentage and then write it as a fraction out of 100. Then simplify the fraction into its simplest form, 18 out of 25, 18/25.

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Lets look in more detail at changing a percentage to a fraction.

Case study _unit2.9.1 Example: 50% is

As you can see this percentage is essentially a fraction of 100. However you can simplify it to .

To change a percentage to a fraction, put the percentage over 100 and simplify if possible.

Sometimes we might see a percentage like this: 12.5%.

If we use the method above we get but we can’t have a decimal in a fraction.

To get rid of the decimal in the fraction we must multiply the top and bottom of the fraction, the numerator and denominator, by any number that will give us whole numbers. In this case 10 or 2 both work well (12.5 × 10 = 125 and 12.5 × 2 = 25):

Extract _unit2.9.1 Method 1: × 10

× top and bottom by 10 = =

Extract _unit2.9.2 Method 2: × 2

× top and bottom by 2 = =

Activity _unit2.9.1 Activity 21: Converting between percentages, decimals and fractions

Express these percentages as decimals:
- a.62%
- b.50%
- c.5%
Express these decimals as percentages:
- a.0.02
- b.0.2
- c.0.752
- d.0.055
Express these percentages as fractions:
- a.15%
- b.2.5%
- c.37.5%

Answer

- a.0.62
- b.0.5
- c.0.05
- a.2%
- b.20%
- c.75.2%
- d.5.5%
- a. =
- b. × top and bottom by 10 = =
- c. × top and bottom by 10 = =
  
  You may have multiplied by different numbers to get rid of the decimal in the last two questions. However, your final answers should still be the same as ours.

Now have a go at matching these fractions to decimals and percentages.

Activity _unit2.9.2 Activity 22: Matching fractions, decimals and percentages

Choose the correct fraction for each percentage and decimal.

Using the following two lists, match each numbered item with the correct letter.

a. 62.5% = 0.625 =
b. 8% = 0.08 =
c. 40% = 0.4 =
d. 35% = 0.35 =

The correct answers are:

1 = d
2 = c
3 = b
4 = a

Next you’ll look in more detail at how to change a fraction to a percentage.

9.2 Changing a fraction to a percentage

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

There are two ways you can do this.

Method 1

To change a fraction into a percentage, multiply it by (essentially, you are just multiplying the top number by 100 and the bottom number will stay the same).

Case study _unit2.9.2 Example: Change into a percentage

Extract _unit2.9.3

× =

This cancels to = 75%

Box _unit2.9.1

Note: Remember anything over 1 is a whole number. If you do not end up with a 1 on the bottom, you will have to divide the top number by the bottom one to get your final answer.

Method 2

Divide the top of the fraction by the bottom (to express the fraction as a decimal) and then multiply the answer by 100.

Case study _unit2.9.3 Example: = 3 ÷ 4 = 0.75

Figure _unit2.9.2 Figure 12 Expressed as a decimal: 3 ÷ 4

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3.00 ÷ 4 = 0.75. This illustration shows that the sum has been handrawn and there is a small ‘3’ above the first ‘0’ of ‘3.00’ and a small ‘2’ above the second ‘0’of 3.00.

Figure 12 Expressed as a decimal: 3 ÷ 4

0.75 × 100 = 75%

Activity _unit2.9.3 Activity 22: Changing a fraction to a percentage

Express these fractions as percentages:
- a.
- b.
- c.

Answer

- a.37.5%
- b.90%
- c.80%

Now you’ll look at changing a fraction to a decimal.

9.3 Changing a fraction to a decimal

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Again there are two ways to do this, both based on the two methods just shown for changing a fraction to a percentage.

Method 1

Case study _unit2.9.4 Example: Change the fraction into a percentage and divide by 100

× = which cancels to = 25%

Now convert to a decimal by dividing by 100:

25 ÷ 100 = 0.25

Method 2

Case study _unit2.9.5 Example: Divide the top of the fraction by the bottom

= 1 ÷ 4 = 0.25.

Figure _unit2.9.3 Figure 13 Expressed as a decimal: 1 ÷ 4

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1.00 ÷ 4 = 0.25. This illustration shows that the sum has been handrawn and there is a small ‘1’ above the first ‘0’ of ‘1.00’ and a small ‘2’ above the second ‘0’of 1.00.

Figure 13 Expressed as a decimal: 1 ÷ 4

Activity _unit2.9.4 Activity 23: Changing a fraction to a decimal

Express these fractions as decimals:

Answer

0.4
0.125
0.3

Fractions and percentages deal with splitting numbers into a given number of equal portions, or parts. When dealing with the next topic, ratio, you will still be splitting quantities into a given number of parts, but when sharing in a ratio, you do not share evenly. This might sound a little complicated but you’ll have been doing it since you were a child.

Summary

In this section you have:

learned about the relationship between fractions, decimals and percentages and are now able to convert between the three.

10 Ratio

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

As you can see from Figure 14, ratio is an important part of everyday life.

Figure _unit2.10.1 Figure 14 Day-to-day ratio

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Cartoon of a teacher and several children playing in a school. A sign reads ‘Adult to child ratio must be 1:4’.

Figure 14 Day-to-day ratio

It is important to understand how to tell which part of the ratio is which. If for example, you have a group of men and women in the ratio of 5:4, as the men were mentioned first, they are the first part of the ratio.

The order of the ratio is very important. Consider the following:

Extract _unit2.10.1

Julia attends a drama club where 100 members are men and 150 members are women. What is the ratio of women to men at the drama club?

Notice how the information that you need to answer the question is given in the opposite order to that required in the answer. It is very important that you give the parts of the ratio in the correct order.

The ratio of women to men is 150:100

If you were asked for the ratio of men to women it would be 100:150

10.1 Simplifying ratios

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Sometimes you need to work out the ratio from the quantities you have.

If we refer back to the example we discussed earlier, we said that the ratio of women to men at the drama club is 150:100. However, you can simplify this ratio by dividing all parts by the same number. This is similar to simplifying fractions, which you have done.

With 150:100, we can divide each side of the ratio by 50 (you could also divide by 10 and then by 5), so the ratio will simplify to 3:2. Therefore, the ratio of women to men at the club is 3:2. Having it written in its simplest form makes it easier to think about and to use for other calculations. For every 2 men you have, there are 3 women.

Let’s look at another example.

Case study _unit2.10.1 Example: Recipes and ratio

Look at this recipe for a mocktail:

Sunset Smoothie

50 ml grenadine
100 ml orange juice
150 ml lemonade

The ratio of the ingredients is:

grenadine:orange juice:lemonade
50 : 100 : 150

To simplify this ratio you can divide all of the numbers by 50 (or by 10 and then 5).

This gives the ratio of grenadine to orange juice to lemonade as 1:2:3.

Activity _unit2.10.1 Activity 24: Simplifying ratios

Simplify the following ratios:

The ratio of women to men in a class is 15:20.
The ratio of management to staff in a warehouse is 10:250.
The ratio of home to away supporters is 24 000 to 8000.
The ratio of votes in a local election was candidate A 1600, candidate B 800, Candidate C 1200.
The ratio of fruit in a bag of mixed dried fruit is 150 g currants, 100 g raisins, 200 g sultanas and 50 g mixed peel.

Answer

Women to men is 3:4 (divide both sides by 5).
Management to staff is 1:25 (divide both sides by 10).
Home to away supporters is 3:1 (divide both sides by 8000 or by 1000 and then by 8).
A to B to C is 4:2:3 (divide each part of the ratio by 400 or by 100 and then by 4).
Currants to raisins to sultanas to mixed peel is 3:2:4:1 (divide by 50 or by 10 and then 5).

Ratio questions can be asked in different ways. There are three main ways of asking a ratio question. Take a look at an example of each below and see if you can identify the differences.

Case study _unit2.10.2 Type 1

A recipe for bread says that flour and water must be used in the ratio 5:3. If you wish to make 500 g of bread, how much flour should you use?

Case study _unit2.10.3 Type 2

You are growing tomatoes. The instructions on the tomato feed say ‘Use 1 part feed to 4 parts water’. If you use 600 ml of water, how much tomato feed should you use?

Case study _unit2.10.4 Type 3

Ishmal and Ailia have shared some money in the ratio 3:7. Ailia receives £20 more than Ishmal. How much does Ishmal receive?

In questions of type 1, you are given the total amount that both ingredients must add to, in this example, 500 g. In questions of type 2 however, you are not given the total amount but instead are given the amount of one part of the ratio. In this case you know that the 4 parts of water total 600 ml.

The final type of ratio question does not give us either the total amount or the amount of one part of the ratio. Instead, it gives us just the difference between the first and second part of the ratio. Whilst neither type of ratio question is more complicated than the others, it is useful to know which type you are dealing with as the approach for solving each type of problem is slightly different.

10.2 Solving ratio problems where the total is given

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

The best way for you to understand how to solve these problems is to look through the worked example in the video below.

Download this video clip.Video player: bltl_1_7_1_ratio.mp4

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Transcript

To understand how to solve ratio problems where the total is given, it's best to look through a worked example.

An art shop is ordering paint. They work out that they need to order red and blue paint in the ratio 3:4. They order a total of 56 cans of paint. How many of those cans are blue paint?

The first step to finding out, is to work out the total number of parts by adding together the parts of the ratio. If there are three parts red and four parts blue, that makes a total of seven parts altogether. Next, you need to work out what one part is worth. To do this, divide the total number of paint cans ordered, by the total number of parts. 56 divided by 7 equals 8. One part is worth eight cans.

Now, you know that one part of paint is 8 cans, you can work out what four parts are worth. 8 x 4 = 32. So the art shop needs to order 32 cans of blue paint.

Can you think of an extra check that you can carry out to confirm your answer? Well as you know what one part of paint is worth, you can also work out how many cans of red paint need to be ordered. 3 parts x 8 cans = 24 cans of red paint.

The total of red paint and blue paint should be 56. So you can check your answer by adding together the number of cans of each. 32 + 24 is indeed 56 cans.

So in summary, the steps for solving a ratio question like this are:

One. Add together the parts of the ratio. 3 + 4 = 7.

Two. Take the total amount given and divide by the sum of the ratio parts. 56 divided by 7 = 8.

Three. Finally, take the answer for step two, 8, and multiply by whichever part of the ratio you're interested in finding. 8 x 4 = 32.

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Activity _unit2.10.2 Activity 25: Ratio problems where the total is known

Try solving these ratio problems:

To make mortar you need to mix soft sand and cement in the ratio 4:1. You need to make a total of 1500 g of mortar.
How much soft sand will you need?

Answer

Add the parts of the ratio:
- 4 + 1 = 5
Divide the total amount required by the sum of the parts of the ratio:
- 1500 g ÷ 5 = 300 g
Since soft sand is 4 parts, we do 300 g × 4 = 1200 g of soft sand.
Check by working out how much cement you need. Cement is 1 part so you would need 300 g:
- 1200 g + 300 g = 1500 g which is the correct total.

To make the mocktail ‘Sea Breeze’ you need to mix cranberry juice and grapefruit juice in the ratio 4:2.
You want to make a total of 2700 ml of mocktail. How much grapefruit juice should you use?

Answer

Add the parts of the ratio:
- 4 + 2 = 6
Divide the total amount required by the sum of the parts of the ratio:
- 2700 ml ÷ 6 = 450 ml
Since grapefruit juice is 2 parts, we do 450 ml × 2 = 900 ml of grapefruit juice.
Check by working out how much cranberry juice you would use:
- 4 × 450 = 1800
  1800 ml + 900 ml = 2700 ml
  You may have simplified the ratio to 2:1 before doing the calculation, but you will see that your answers are the same as ours.

The instructions to mix Misty Morning paint are mix 150 ml of azure with 100 ml of light grey and 250 ml of white paint.
How much light grey paint would you need to make 5 litres of Misty Morning?

Answer

Start by expressing and then simplifying the ratio:
- 150:100:250 which simplifies to 3:2:5 = 10 parts
- 5 litres = 5000 ml (converting to ml makes your calculation easier.)
Divide the total amount required by the sum of the parts of the ratio:
- 5000 ÷ 10 = 500 so 1 part = 500 ml
- Light grey is 2 parts:
  2 × 500 = 1000 ml or 1 litre
Check:
- azure is 3 parts: 3 × 500 = 1500 ml or 1.5 litres
- white is 5 parts: 5 × 500 = 2500 ml or 2.5 litres
- 1000 + 1500 + 2500 = 5000 ml or 5 litres

You want to make 14 litres of squash for a children’s party. The concentrate label says mix with water in the ratio of 2:5.
How much concentrate will you use?

Answer

Add the parts of the ratio:
- 2 + 5 = 7
Divide the total amount required by the sum of the parts of the ratio:
- 14 litres ÷ 7 = 2 litres so 1 part = 2 litres
(Note: this calculation was straightforward so there was no need to convert to ml.)
Since the concentrate is 2 parts you will need 2 litres × 2 = 4 litres of concentrate.
Check:
- Water is 5 parts:
- 5 × 2 litres = 10 litres
- 4 + 10 = 14 litres.

A man leaves £8400 in his will to be split between 3 charities:
- Dogs Trust, RNLI and MacMillan Research in the ratio 3:2:1.
How much will each charity receive?

Answer

Add the parts of the ratio:
- 3 + 2 + 1 = 6
Divide the total amount required by the sum of the parts of the ratio:
- £8400 ÷ 6 = 1400
- – The Dogs Trust receives 3 parts: 3 × £1400 = £4200
- – The RNLI receives 2 parts: 2 × £1400 = £2800
- – MacMillan Research receives 1 part so: £1400
- Check:
  4200 + 2800 + 1400 = £8400

Next you’ll look at ratio problems where the total of one part of the ratio is known.

10.3 Solving ratio problems where the total of one part of the ratio is given

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Take a look at the worked example below:

Extract _unit2.10.2

You are growing tomatoes. The instructions on the tomato feed say:

Use 1 part feed to 4 parts water

If you use 600 ml of water, how much tomato feed should you use?

These questions make much more sense if you look at them visually:

Extract _unit2.10.3

Figure _unit2.10.2 Figure 15 Solving ratio problems to grow tomatoes

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Illustration showing a jug filled up to 600 ml with water. The text reads ‘Feed:Water, 1:4’. An arrow points from the number 4 to ‘600 ml’. Another arrow points from the number 1 to a question mark.

Figure 15 Solving ratio problems to grow tomatoes

Extract _unit2.10.4

You can now see clearly that 600 ml of water is worth 4 parts of the ratio. To find one part of the ratio you need to do:

600 ml ÷ 4 = 150 ml

Since the feed is only 1 part, feed must be 150 ml. If feed was more than one part you would multiply 150 ml by the number of parts.

Just as with the previous type of question, you need to try to work out the value of 1 part. The value of any other number of parts can be worked out from this.

Activity _unit2.10.3 Activity 26: Ratio problems with one part given

Practise your skills by tackling the ratio problems below:

A recipe requires flour and butter to be used in the ratio 3:5. The amount of butter used is 700 g.
How much flour will be needed?

Answer

Flour:Butter
Figure _unit2.10.3 Figure 16 Using ratios in recipes
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Illustration showing kitchen weighing scales measuring 700 g. The text reads ‘Flour:Butter, 3:5’. An arrow points from the number 5 to ‘700 g’. Another arrow points from the number 3 to a question mark
Figure 16 Using ratios in recipes
To find one part you do 700 g ÷ 5 = 140 g
To find the amount of flour needed you then do 140 g × 3 = 420 g flour.

When looking after children aged between 7 and 10, the ratio of adults to children must be 1:8.
- a.For a group of 32 children, how many adults must there be?
- b.If there was one more child in the group, how would this affect the number of adults required?

Answer

Adults:Children
- a.
  Figure _unit2.10.4 Figure 17 Working out the ratio of adults to children
  Show description|Hide description
  Illustration showing an adult and 8 children. The text reads ‘Adults:Children, 1:8’. An arrow points from the number 8 to ‘32’. Another arrow points from the number 1 to a question mark.
  Figure 17 Working out the ratio of adults to children
  To find one part you do 32 ÷ 8 = 4.
  Since adults are only 1 part, you need 4 adults.
- b.If there were 33 children, one part would be 33 ÷ 8 = 4.125.
  Since you cannot have 4.125 adults, you need to round up to 5 adults so you would need one more adult for 33 children.

A shop mixes bags of muesli using oats, sultanas and nuts in the ratio 6:3:1.
If the amount of sultanas used is 210 g, how heavy will the bag of muesli be?

Answer

Oats:Sultanas:Nuts
Figure _unit2.10.5 Figure 18 Working out the ratio of oats, sultanas and nuts
Sultanas are 3 parts so to find 1 part you do 210 g ÷ 3 = 70 g.
Oats are 6 parts so 6 × 70 = 420 g.
Nuts are only 1 part so they are 70 g.
The total weight of the bag would be 210 g + 420 g + 70 g = 700 g.

Next you’ll look at ratio problems where only the difference in amounts is given.

10.4 Solving ratio problems where only the difference in amounts is given

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Earlier in the section you came across the question below. Let’s have a look at how we could solve this.

Case study _unit2.10.5 Example: Solving ratio amounts from the difference

Ishmal and Ailia have shared some money in the ratio 3:7.

Ailia receives £20 more than Ishmal. How much does Ishmal receive?

Ishmal:Ailia
3:7

You know that the difference between the amount received by Ishmal and the amount received by Ailia is £20. You can also see that Ailia gets 7 parts of the money whereas Ishmal only gets 3.

The difference in parts is therefore 7 − 3 = 4. So 4 parts = £20.

Now this is established, you can work out the value of one part by doing:

£20 ÷ 4 = £5

As you want to know how much Ishmal received you now do:

£5 × 3 = £15

As an extra check, you can work out Ailia’s by doing:

£5 × 7 = £35

This is indeed £20 more than Ishmal.

Activity _unit2.10.4 Activity 27: Ratio problems where difference given

Now try solving this type of problem for yourself.

The ratio of female to male engineers in a company is 2:9. At the same company, there are 42 more male engineers than females.
How many females work for this company?
A garden patio uses grey and white slabs in the ratio 3:5. You order 30 fewer grey slabs than white slabs.
How many slabs did you order in total?

Answer

The difference in parts between males and females is 9 − 2 = 7 parts.
You know that these 7 parts = 42 people.
To find 1 part you do:
- 42 ÷ 7 = 6
Now you know that 1 part is worth 6 people, you can find the number of females by doing
- 6 × 2 = 12 females
Check:
- The number of males is 6 × 9 = 54.
  The difference between 54 and 12 is 42.
The difference in parts between grey and white is 5 − 3 = 2 parts.
These 2 parts are worth 30. To find 1 part you do:
- 30 ÷ 2 = 15
To find grey slabs do:
- 15 × 3 = 45
To find white slabs do:
- 15 × 5 = 75
Check:
- The difference between the number of grey and white slabs is 30 (75 − 45).
  Now you know both grey and white totals, you can find the total number of slabs by doing:
- 45 + 75 = 120 slabs in total.

Even though there are different ways of asking ratio questions, the aim of any ratio question is to determine the value of one part. Once you know this, the answer is simple to find!

Ratio can also be used in less obvious ways. Imagine you are baking a batch of scones and the recipe makes 12 scones. However, you need to make 18 scones rather than 12. How do you work out how much of each ingredient you need? The final ratio section deals with other applications of ratio.

10.5 Other applications of ratio

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

A very common and practical use of ratio is when you want to change the proportions of a recipe. All recipes state the number of portions they will make, but this is not always the number that you wish to make.You may wish to make more or less than the actual recipe gives. If you wanted to make 18 scones but only have a recipe that makes 12, how do you know how much of each ingredient to use?

To make 12 scones
400 g self-raising flour
1 tablespoon caster sugar
80 g butter
250 ml milk

Figure _unit2.10.6 Figure 19 Scones on a plate

As you already know the ingredients to make 12 scones, you need to know how much of each ingredient to make an extra 6 scones. Since 6 is half of 12, if you halve each ingredient, you will have the ingredients for the extra 6 scones. To find the total for 18 scones you need to add together the ingredients for the 12 scones and the 6 scones.

Table _unit2.10.1 Table 9
12 scones	6 scones	18 scones
400 g flour	400 g ÷ 2 = 200 g flour	400 g + 200 g = 600 g flour
1 tablespoon caster sugar	1 ÷ 2 = tablespoon caster sugar	1 + = 1 tablespoons caster sugar
80 g butter	80 g ÷ 2 = 40 g butter	80 g + 40 g = 120 g butter
250 ml milk	250 ml ÷ 2 = 125 ml milk	250 ml + 125 ml = 375 ml milk

Have a go at the activity below to check your skills.

Activity _unit2.10.5 Activity 28: Ratio and recipes

This recipe makes 18 biscuits:
- 220 g self-raising flour
- 150 g butter
- 100 g caster sugar
- 2 eggs

How much of each ingredient is needed for 9 biscuits?

Answer

Since 9 is half of 18, you need to halve each ingredient to find the amount required to make 9 biscuits.
- 220g ÷ 2 = 110 g flour
- 150g ÷ 2 = 75 g butter
- 100g ÷ 2 = 50 g sugar
- 2 ÷ 2 = 1 egg

To make strawberry milkshake you need:
- 630 ml milk
- 3 scoops of ice cream
- 240 g of strawberries
- The recipe serves 3

How much of each ingredient is needed for 9 people?

Answer

You know the ingredients for 3 but want to know the ingredients for 9. Since 9 is three times as big as 3, you need to multiply each ingredient by 3.
- 630 ml × 3 = 1890 ml milk
- 3 × 3 = 9 scoops of ice cream
- 240 g × 3 = 720 g of strawberries

Angel Delight recipe:
- Add 60 g powder to 300 ml cold milk
- Serves 2 people

How much of each ingredient is needed to serve 5 people?

Answer

You could work this out in 2 different ways.
Method 1
You know the ingredients for 2 people. You can find ingredients for 4 people by doubling the ingredients for 2. You then need ingredients for an extra 1 person. Since 1 is half of 2, you can halve the ingredients for 2 people.
- 60 g + 60 g + 30 g = 150 g powder
- 300 ml + 300 ml + 150 ml = 750 ml milk
Method 2
You know the ingredients for 2 people so you can find the ingredients for 1 person by halving them. You can then multiply the ingredients for 1 person by 5.
- 60g ÷ 2 = 30 × 5 = 150 g powder
- 300ml ÷ 2 = 150 × 5 = 750 ml milk

The final practical application of ratio can be very useful when you are out shopping. Supermarkets often try and encourage us to buy in bulk by offering larger ‘value’ packs. But how can you work out if this is actually a good deal? Take a look at the example below.

Case study _unit2.10.6 Example: Ratio and shopping

Which of the boxes below offers the best value for money?

Figure _unit2.10.7 Figure 20 Shopping options: tea

Show description|Hide description

Illustration of two boxes of teabags. One box contains 40 teabags and costs £1.20. The other box contains 240 teabags and costs £9.60.

Figure 20 Shopping options: tea

There are various ways of comparing the prices.

Method 1

To work out which is the best value for money we need to find the price of 1 teabag.

If 40 teabags cost £1.20 then to find the cost of 1 teabag you do:

Extract _unit2.10.5

£1.20 ÷ 40 = £0.03, or 3p

If 240 teabags cost £9.60 then to find the cost of 1 teabag you do:

Extract _unit2.10.6

£9.60 ÷ 240 = £0.04, or 4p

The box containing 40 teabags is therefore better value than the larger box.

Method 2

The ratio of teabags is

Extract _unit2.10.7

40 : 240 which you can simplify to 1:6

If we use the price for the small box you can see that 1 part is £1.20

You can then use this value to calculate the price of the large box. At this price, the bigger box would be £1.20 × 6 = £7.20 so we can see that the small box is better value.

Activity _unit2.10.6 Activity 29: Practical applications of ratio

Use whichever method you prefer to work out the best deal in each case.

Work out which deal is the best value for money.
- a.
  Figure _unit2.10.8 Figure 21 Cola options
  Show description|Hide description
  Two bottles of cola. One is 2 litres and costs 64 pence. The other is 3 litres and costs 99 pence.
  Figure 21 Cola options

Answer

- a.2 litres cost 64 p, so 1 litre costs 64 p ÷ 2 = 32p.
  3 litres cost 99p, so 1 litre costs 99p ÷ 3 = 33p.
  Comparing the cost of 1 litre in each case, we see that the 2-litre bottle is the best buy.

- b.
  Figure _unit2.10.9 Figure 22 Milk options
  Show description|Hide description
  Two cartons of milk. One is 1 pint and costs 26 pence. The other is 4 pints and costs 92 pence.
  Figure 22 Milk options

Answer

- b.1 pint costs 26p.
  4-pint carton costs 92p, so 1 pint costs 92p ÷ 4 = 23p.
  Comparing the cost of 1 pint of milk in each case, we see that the 4-pint carton is the best buy.

- c.
  Figure _unit2.10.10 Figure 23 Washing powder options
  Show description|Hide description
  Two boxes of washing powder. One is 2 kg and costs £3. The other is 5 kg and costs £10.
  Figure 23 Washing powder options

Answer

- c.5 kg costs £10, so 1 kg costs £10 ÷ 5 = £2.
  2 kg cost £3, so 1 kg costs £3 ÷ 2 = £1.50.
  Comparing the cost of 1 kg of powder in each case, we see that the 2 kg box is the best buy.

Two supermarkets sell the same brand of juice. Shop B is offering ‘buy one get second one half price’ for apple juice and ‘buy one get one free’ for orange juice.
For each type of juice which shop is offering the best deal?
- a.
  Figure _unit2.10.11 Figure 24 Apple juice options
  Show description|Hide description
  Two 1 litre cartons of apple juice. The cost in Shop A is 52 pence. The cost in Shop B is 72 pence plus the offer ‘Buy one get second one half price’.
  Figure 24 Apple juice options

Answer

- a.Shop A: 1 litre costs 52p
  Shop B: 2 litres cost 72p + 36p = 108p (here we pay 72p for the first litre and 36p for second litre), so 1 litre costs 108p ÷ 2 = 54p.
  Comparing the cost of 1 litre of apple juice in each case, we see that Shop A offers the better deal.

- b.
  Figure _unit2.10.12 Figure 25 Orange juice options
  Show description|Hide description
  Two 1 litre cartons of orange juice. The cost in Shop A is 39 pence. The cost in Shop B is 76 pence plus the offer ‘Buy one get one free’.
  Figure 25 Orange juice options

Answer

- b.Shop A: 1 litre costs 39p
  Shop B: 2 litres cost 76p (we get 1 litre free), so 1 litre costs: 76p ÷ 2 = 38p.
  Comparing the cost of 1 litre of orange juice in each case, we see that Shop B offers the better deal.

A supermarket sells bread rolls in 3 different size packs. Which size offers the best value for money?
Figure _unit2.10.13 Figure 26 Bread rolls options
Show description|Hide description
Three packs of bread rolls: ‘18 rolls £3.42’; 12 rolls £2.16; ‘4 rolls 80p’.
Figure 26 Bread rolls options

Answer

Calculate the cost of 1 roll in each pack:
- 80p ÷ 4 = 20p
  £2.16 ÷ 12 = £0.18 or 18p
  £3.42 ÷ 18 = £0.19 or 19p
The pack of 12 is best value.

You have now completed all elements of the ratio section and hopefully are feeling confident with each topic.

The next section of the course deals with formulas. This might sound daunting but you have actually already used a formula. Remember when you learned about how to work out the percentage change of an item? To do that you used a simple formula and you will now take a closer look at slightly more complex formulas.

Summary

In this section you have:

learned about the three different types of ratio problems and that the aim of any ratio problem is to find out how much one part is worth
practised solving each type of ratio problem:
- where the total amount is given
- where you are given the total of only one part
- where only the difference in amounts is given
learned about other useful applications of ratio, such as changing the proportions of a recipe.

11 Formulas

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Figure _unit2.11.1 Figure 27 Formulas

Show description|Hide description

Illustration showing examples of formulas.

Figure 27 Formulas

Before diving in to this topic, you first need to learn about the order in which you need to carry out addition, subtraction, multiplication and division. Have you ever seen a question like the one below posted on social media?

Figure _unit2.11.2 Figure 28 A calculation using the four operations

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Illustration showing the text ‘What is the answer?’ and the sum ‘7 + 7 ÷ 7 + 7 x 7 – 7’.

Figure 28 A calculation using the four operations

There are usually a wide variety of answers given by various people. But how is it possible that such a simple calculation could cause so much confusion? It’s all to do with the order in which you carry out the calculations.

If you go from left to right:
- 7 + 7 = 14
- 14 ÷ 7 = 2
- 2 + 7 = 9
- 9 × 7 = 63
- 63 − 7 = 56

Check this on a calculator and you will see that the correct answer is actually 50. How do you arrive at this answer? You have to use the correct order of operations, sometimes called BIDMAS.

11.1 Order of operations

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

The order in which you carry out operations can make a big difference to the final answer. When doing any calculation that involves doing more than one operation, you must follow the rules of BIDMAS in order to arrive at the correct answer.

Figure _unit2.11.3 Figure 29 The BIDMAS order of operations

Show description|Hide description

Illustration of the word BIDMAS. ‘B’ stands for Brackets. ‘I’ stands for Indices. ‘DM’ stands for Divide & Multiply. ‘AS’ stands for Add & Subtract. An arrow under the word BIDMAS is labelled ‘Order of Operations’ and points from left to right.

Figure 29 The BIDMAS order of operations

Case study _unit2.11.1 B: Brackets

Any calculation that is in brackets must be done first.

Example:

Extract _unit2.11.1

2 × (3 + 5)

2 × 8 = 16

Note that this could also be written as 2 (3 + 5) because if a number is next to a bracket, it means you need to multiply.

If there is more than 1 operation in the brackets, you must follow the rules of BIDMAS in the brackets.

Case study _unit2.11.2 I: Indices

After any calculations in brackets have been done, you must deal with any calculations involving indices or powers i.e.

Extract _unit2.11.2

3² = 3 × 3

4³ = 4 × 4 × 4

Example:

Extract _unit2.11.3

3 × 4²

3 × (4 × 4)

3 × 16 = 48

Case study _unit2.11.3 D: Divide

Next come any division or multiplication calculations. Of these two calculations, you should do them in the order that they appear in the sum from left to right.

Example:

Extract _unit2.11.4

16 − 10 ÷ 5

16 − 2 = 14

Case study _unit2.11.4 M: Multiply

Example:

Extract _unit2.11.5

5 + 6 × 2

5 + 12 = 17

Case study _unit2.11.5 A: Add

Finally, any calculations involving addition or subtraction are done. Again, these should be done in the order that they appear from left to right.

Case study _unit2.11.6 S: Subtract

Example:

Extract _unit2.11.6

24 + 10 − 2

34 − 2 = 32

24 + 8 = 32

Activity _unit2.11.1 Activity 30: Using BIDMAS

Now have a go at carrying out the following calculations yourself. Make sure you apply BIDMAS!

4 + 3 × 2
5 (4 − 1)
36 ÷ 3²
7 + 15 ÷ 3 − 4

Answer

4 + 6 = 10
5 × 3 = 15
36 ÷ 9 = 4
7 + 5 − 4 = 8

Now that you have learned the rules of BIDMAS you are ready to apply them when using formulas.

11.2 Formulas in practice

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

You will already have come across and used formulas in your everyday life. For example, if you are trying to work out the cost of a new carpet you will have used the formula:

area = length × width

to calculate how much carpet you would need.

Often division in a formula is shown as one number over another, for example:

6 ÷ 3 would be shown as

Let’s look at division in a formula:

Extract _unit2.11.7

A lorry driver travels 120 miles in 3 hours. What was the average speed during the journey?

Box _unit2.11.1

Note: As the lorry driver was unlikely to have travelled at a constant speed for 120 miles we say we are calculating the average speed as this will give us the typical overall speed.

speed =
speed = 40 miles per hour

Sometimes we use letters to represent the different elements used in a formula, e.g. the formula above might be shown as:

s =
where:
- ‘s’ = speed in mph
- ‘d’ = distance in miles
- ‘t’ = time in hours

If you are trying to work out the time to cook a fresh chicken you may have used the formula:

Extract _unit2.11.8

Time (minutes) = 15 + × 25 where ‘w’ is the weight of the chicken in grams.

For example, if you wanted to cook a chicken that weighs 2500 g you would do:

Time (minutes) = 15 + × 25

Remembering to use BIDMAS you would then get:

Time (minutes) = 15 + 5 × 25
= 15 + 125
= 140 minutes

Let’s look at another worked example before you try some on your own.

Case study _unit2.11.7 Example: Gas bill formula

Extract _unit2.11.9

The owner of a guesthouse receives a gas bill. It has been calculated using the formula:

Cost of gas (£) =

Box _unit2.11.2

Note: 8d means you do 8 × d.

Where d = number of days and u = number of units used, if she used 3500 units of gas in 90 days, how much is the bill?

In this example, d = 90 and u = 3500 so you do:

Extract _unit2.11.10

Cost of gas (£) =

= £42.20

Activity _unit2.11.2 Activity 31: Using formulas

Fuel consumption in Europe is calculated in litres per 100 kilometres. A formula to approximate converting from miles per gallon to litres per 100 kilometres is:
- L =
where L = number of litres per 100 kilometres and M = number of miles per gallon.
A car travels 40 miles per gallon. What is this in litres per kilometres?

Answer

- L = and in this case M = 40
- L =
- L = 7 litres per 100 kilometres

Using the formula I = where:
- I = interest
- P = principal amount of loan
- R = interest rate
- T = time in years
calculate how much interest is due on a loan of £5000 taken over 3 years at an interest rate of 5.5%.

Answer

I =
In this case P = £5000, R = 5.5% and T = 3 years.

I =

I =

I = 825
So the interest paid would be £825.

The area of a trapezium can be calculated using the formula:
- A =
Figure _unit2.11.4 Figure 30 Dimensions of a trapezium
Show description|Hide description
A trapezium. The length of the top side is labelled ‘a’. The length of the bottom side is labelled ‘b’. The length between the top and bottom sides (height) is labelled ‘h’.
Figure 30 Dimensions of a trapezium
Find the area of trapeziums where:
- i.a = 5 cm, b = 9 cm and h = 7 cm
- ii.a = 35 mm, b = 40 mm and h = 10 cm

Answer

A =
- i.A =
  
  A =
  
  A =
  
  A = 49 cm²
- ii.In this question you must convert the units so that they are all the same. The units that you select will be the units that your answer will be given in, e.g. if you convert to mm your answer will be in mm² but if you convert to cm your answer will be in cm².
  
        A =
  
  Method 1 – converting to mm
  Convert h measurement to mm:
        10 × 10 = 100 mm
  
        A =
  
        A =
  
        A =
  
        A = 3750 mm²
  
  Method 2 – converting to cm
  Convert a and b measurements to cm:
        a = 35 ÷ 10 = 3.5 cm
        b = 40 ÷ 10 = 4 cm
  
        A =
  
        A =
  
        A =
  
        A = 37.5 cm²

Box _unit2.11.3

Note: it is a good idea to show all the stages of the calculation to help you keep track of your workings.

A company uses the following formula to work out the total cost to the customer of hiring a bouncy castle:
- T = hc + (0.45d) + 15
  where:
        T = total
        h = number of days hire
        c = cost of castle per day
        d = delivery distance in miles.
Figure _unit2.11.5 Figure 31 Dues’s Bouncy Fun – price list
Show description|Hide description
Illustration of an advert for ‘Duey’s Bouncy Fun’ which is displayed at the top. A picture of a bouncy castle is on the left. On the right is the price list: Superhero Castle £34.50; Supersonic Castle £42.00; Stars and Unicorns Palace £45.00.
Figure 31 Dues’s Bouncy Fun – price list
Stuart lives 12 miles away and would like to hire a Supersonic Castle for 2 days. How much will it cost?

Answer

T = hc + (0.45d) + 15
In this case h = 2, c = £42, and d = 12, so:
      T = 2 × 42 + (0.45 × 12) + 15
      T = 84 + 5.4 + 15
      T = 104.4
The total cost of hire would be £104.40.

Now that you have learned all the skills that relate to the number section of this course, there is just one final thing you need to be able to do before you will be ready to complete the end-of-session quiz for numbers.

You are now proficient at carrying out lots of different calculations including working out fractions and percentages of numbers, using ratio in different contexts and using formulas.

It is fantastic that you can now do all these things, but how do you check if an answer is correct? One way you can check would be to approximate an answer to the calculation (as you did in Section 3.2). Another way to check an answer is to use the inverse (opposite) operation.

Summary

In this section you have:

learned about, and practised using BIDMAS – the order in which operations must be carried out
seen examples of formulas used in everyday life and practised using formulas to solve a problem.

12 Checking your answers

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Figure _unit2.12.1 Figure 32 Inverse operations

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A clapper board with the title ‘Inverse Operations’ and showing a two-row table. The first row of the table reads ‘Operation: +, –, x, ÷’. The second row of the table reads ‘Inverse: –, +, ÷, x’.

Figure 32 Inverse operations

An inverse operation is an opposite operation. In a sense, it ‘undoes’ the operation that has just been performed. Let’s look at two simple examples to begin with.

Case study _unit2.12.1 Example: Check your working 1

6 + 10 = 16

Method

Since you have done an addition sum, the inverse operation is subtraction. To check this calculation, you can either do:

Extract _unit2.12.1

16 − 10 = 6

16 − 6 = 10

You will notice here that the same 3 numbers (6, 10 and 16) have been used in all the calculations.

Case study _unit2.12.2 Example: Check your working 2

5 × 3 = 15

Method

This time, since you have done a multiplication sum, the inverse operation is division. To check this calculation, you can either do:

Extract _unit2.12.2

15 ÷ 5 = 3

15 ÷ 3 = 5

Again, you will notice that the same 3 numbers (3, 5 and 15) have been used in all the calculations.

If you have done a more complicated calculation, involving more than one step, you simply ‘undo’ each step.

Case study _unit2.12.3 Example: Check your working 3

A coat costing £40 has a discount of 15%. How much do you pay?

Method

Firstly, we find out 15% of £40:

Extract _unit2.12.3

40 ÷ 100 × 15 = £6 discount

£40 − £6 = £34 to pay

To check this calculation, firstly you would check the subtraction sum by doing the addition:

Extract _unit2.12.4

£34 + £6 = £40

To check the percentage calculation you then do:

Extract _unit2.12.5

£6 ÷ 15 × 100 = £40

Don’t forget, sometimes it can also be helpful to use estimation to check your answers, particularly when using decimal or large numbers.

You have now completed the number section of the course. Before moving on to the next session, ‘Units of measure’, complete the quiz on the following page to check your knowledge and understanding.

Summary

In this section you have:

learned that each of the four operations has an inverse operation (its opposite) and that these can be used to check your answers
seen examples showing how to check answers using the inverse operation.

13 Session 1 quiz

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Now it’s time to review your learning in the end-of-session quiz.

Session 1 quiz.

Open the quiz in a new window or tab (by holding ctrl [or cmd on a Mac] when you click the link), then return here when you have done it.

Although the quizzes in this course do not require you to show your working to gain marks, real exams would do so. We therefore encourage you to practise this by using a paper and pen to clearly work out the answers to the questions. This will also help you to make sure you get the right answer.

14 Session 1 summary

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

You have now completed Session 1, ‘Working with numbers’. If you have identified any areas that you need to work on, please ensure you refer back to this section of the course and retry the activities.

You should now be able to:

use the four operations to solve problems in context
understand rounding and look at different ways of doing this
write large numbers in full and shortened forms
carry out calculations with large numbers
carry out multistage calculations
solve problems involving negative numbers
define some key mathematical terms (multiple, lowest common multiple, factor, common factor and prime number)
identify lowest common multiples and factors
use fractions, decimals and percentages and convert between them
solve different types of ratio problems
make substitutions within given formulas to solve problems
use inverse operations and estimations to check your calculations.

All of the skills above will help you with tasks in everyday life. Whether you are at home or at work, number skills are an essential skill to have.

You are now ready to move on to Session 2, ‘Units of measure’.

Introduction

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

In this session you will be calculating using units of measure and focusing on length, time, weight, capacity and money. You will already be using these skills in everyday life when:

working out which train you need to catch to get to your meeting on time
weighing or measuring ingredients when cooking
doing rough conversions if you are abroad to work out how much your meal costs in British pounds.

By the end of this session you will be able to:

understand that there are different units used for measuring and how to choose the appropriate unit
convert between measurements in the same system (e.g. grams and kilograms) and those in different systems (e.g. litres and gallons)
use exchange rates to convert currencies
work with time and timetables
work out the average speed of a journey using a formula
convert temperature measurements between Celsius (°C) and Fahrenheit (°F)
read scales on measuring equipment.

First watch the video below which introduces units of measure.

Download this video clip.Video player: bltl_a68_measure.mp4

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Transcript

[MUSIC PLAYING]

A unit of measure is the unit you use to measure something, whether it's weight, time, volume, or simply measuring the size of something. Knowing the appropriate unit to use in any situation can be very important. And can avoid some basic errors.

Learning to convert between different units of measure can also be very useful. Knowing how to weigh your luggage correctly could even save you money. And understanding the relationship between units of measure, like speed and distance, can be vital if you're planning a journey and can't afford to waste time. But though understanding units of measure is a very useful skill, it can't help in every situation.

End transcript

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1 Units of measure

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

A unit of measure is simply what you measure something in and you will already be familiar with using centimetres, metres, kilograms, grams, litres and millilitres for measurement. When measuring something, you need to choose the appropriate unit of measure for the item you are measuring. You would not, for example, measure the length of a room in millimetres. Have a go at the short activity below and choose the most appropriate unit for each example.

Activity _unit3.1.1 Activity 1: Choosing the unit

Using the following two lists, match each numbered item with the correct letter.

Millilitres (ml)
Metres (m)
Kilograms (kg)
Grams (g)
Centimetres (cm)
Litres (l)

a.Weight of an egg
b.Weight of a dog
c.Amount of water in a paddling pool
d.Amount of liquid in a glass
e.Length of a garden fence
f.Length of a computer screen

The correct answers are:

1 = d
2 = e
3 = b
4 = a
5 = f
6 = c

Hopefully you found that activity fairly straightforward. Next you’ll take a closer look at some units of measure and how to convert between them.

1.1 Converting units of measure in the same system

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Imagine you are catering for a party and go to the wholesale shop to buy flour. You need 200 g of flour for each batch of cookies you will be making and need to make 30 batches. You can buy a 5 kg bag of flour but aren’t sure if that will be enough.

In order to work out if you have enough flour, you need both measurements to be in the same unit – kg or g – before you can do the calculation. For units of measure that are in the same system (so you are not dealing with converting cm and inches for example) it’s a simple process to convert one measurement into another.

For now we will focus on what is known as the metric system of measurement (we will look at some units from the imperial system later on in the session). In most cases, if you want to convert between units in the metric system you will just need to multiply or divide by 10, 100 or 1000.

Take a look at the diagrams below which explain how to convert each type of measurement unit.

Length

Figure _unit3.1.1 Figure 1 A conversion chart for length

Show description|Hide description

A conversion chart for length. cm × 10 for mm; mm ÷ 10 for cm m × 100 for cm; cm ÷ 100 for m km × 1000 for m; m ÷ 1000 for km

Figure 1 A conversion chart for length

Weight

Figure _unit3.1.2 Figure 2 Converting between grams and kilograms

Show description|Hide description

A conversion chart for weight. g ÷ 1000 for kg; kg × 1000 for g

Figure 2 Converting between grams and kilograms

Figure _unit3.1.3 Figure 3 Converting between milligrams and grams

Show description|Hide description

A conversion chart for weight. mg ÷ 1000 for g; g × 1000 for mg.

Figure 3 Converting between milligrams and grams

Money

Figure _unit3.1.4 Figure 4 A conversion chart for money

Show description|Hide description

A conversion chart for money. p ÷ 100 for £; £ × 100 for p

Figure 4 A conversion chart for money

Capacity

Figure _unit3.1.5 Figure 5 Converting between millilitres and litres

Show description|Hide description

A conversion chart for capacity. ml ÷ 1000 for l; l × 1000 for ml

Figure 5 Converting between millilitres and litres

Figure _unit3.1.6 Figure 6 Converting between millilitres and centilitres

Show description|Hide description

A conversion chart for capacity. ml ÷ 10 for cl; cl × 10 for ml

Figure 6 Converting between millilitres and centilitres

Box _unit3.1.1

Note: Capacity may also be measured in centilitres (shortened to cl). On a standard bottle of water, for instance, it may give the capacity as 500 ml or 50 cl – there are 10 millilitres (ml) in 1 centilitre (cl).

When converting between units you may need to do the conversion in stages.

Case study _unit3.1.1 Example: Converting length

You measure the gap for a washing machine to be 0.62 m. You look on a retailer’s website and it shows the width of a particular washing machine to be 600 mm. Will this fit in the gap?

Method

You can convert the gap for the washing machine in stages. Convert from metres to centimetres first by multiplying by 100:

Extract _unit3.1.1

0.62 m × 100 = 62 cm

Now convert the centimetres figure into millimetres by multiplying by 10:

Extract _unit3.1.2

62 cm × 10 = 620 mm

So the washing machine (600 mm wide) will fit in the gap of 620 mm.

You may have preferred to convert the washing machine width into metres instead and do the comparison that way:

Extract _unit3.1.3

600 mm ÷ 10 = 60 cm

60 cm ÷ 100 = 0.6 m

0.6 m is less than 0.62 m so the washing machine will fit the gap.

You’ll learn how to convert between different currencies and units of measure in different systems (metric system kilograms (kg) to imperial system pounds (lb) for example) later in this session. For now, have a go at the activities below and test your metric conversion skills.

Activity _unit3.1.2 Activity 2: Converting units

Use your conversion skills to fill in the missing information in Table 1(a). You may want to carry out some of the calculations in stages.

Please carry out all of your calculations without using a calculator. However, double-check on a calculator if you need to.

Table _unit3.1.1 Table 1(a)
Length	Weight	Capacity
55cm = ? mm	9.75 kg = ? g	235 ml = ? l
257 cm = ? m	652 g = ? kg	18.255 l = ? ml
28.7 km = ? m	5846 g = ? kg	16 ml = ? l
769 mm = ? cm	19.4 kg = ? g	7.88 l = ? ml
1.43 m = ? mm	44 g = ? mg	250 ml = ? cl
125 500 cm = ? km	750 000 mg = ? kg	7.4 l = ? cl

Answer

Table _unit3.1.2 Table 1(b)
Length	Weight	Capacity
55cm × 10 = 550 mm	9.75 kg × 1000 = 9750 g	235 ml ÷ 1000 = 0.235 l
257 cm ÷ 100 = 2.57 m	652 g ÷ 1000 = 0.652 kg	18.255 l × 1000 = 18 225 ml
28.7 km × 1000 = 28 700 m	5846 g ÷ 1000 = 5.846 kg	16 ml ÷ 1000 = 0.016 l
769 mm ÷ 10 = 76.9 cm	19.4 kg × 1000 = 19 400 g	7.88 l × 1000 = 7880 ml
1.43 m × 100 = 143 cm × 10 = 1430 mm	44 g × 1000 = 44 000 mg	250 ml ÷ 10 = 25 cl
125 500 cm ÷ 100 = 1255 m ÷ 1000 = 1.255 km	750 000 mg ÷ 1000 = 750 g ÷ 1000 = 0.75 kg	7.4 l × 100 = 740 cl

Activity _unit3.1.3 Activity 3: Solving conversion problems

Solve the following problems. Please carry out all of your calculations without using a calculator. You may double check on a calculator if needed.

A sunflower is 1.8 m tall. Over the next month, it grows a further 34 cm. How tall is the sunflower at the end of the month?
Peter is a long distance runner. In a training session he runs around the 400 m track 23 times. He wanted to run a distance of 10 km. How many more times does he need to run around the track to achieve this?
A water cooler comes with water containers that hold 12 l of water. The cups provided for use hold 150 ml. A company estimates that each of its 20 employees will drink 2 cups of water a day. How many 12 l bottles will be needed for each working week (Monday to Friday)?
David is a farmer and has 52 goats. He is given a bottle of wormer that contains 0.3 litres. The bottle comes with the instructions:
- ‘Use 4 ml of wormer for each 15 kg of body weight.’
The average weight of David’s goats is 21 kg and he wants to treat all of them. Does David have enough wormer?

Answer

1 m = 100 cm so 1.8 × 100 = 180 cm
180 cm + 34 cm = 214 cm or 2.14 m
400 m × 23 = 9200 m that Peter has run.
1 km = 1000 m so, 10 km = 10 × 1000 = 10 000 m
The difference between what Peter wants to run and what he has already run is:
- 10 000 − 9200 = 800 m
- 800 ÷ 400 = 2
Peter needs to run another 2 laps of the track.
1 litre = 1000 ml so 12 l = 12 × 1000 = 12 000 ml
Each employee will drink 2 × 150 ml = 300 ml a day
There are 20 employees so 300 × 20 = 6000 ml for all employees per day
A working week is 5 days:
- 6000 × 5 = 30 000 ml for all employees for the week
- 30 000 ÷ 12 000 = 2.5
They will need 2 and a half containers per week.
1 litre = 1000 ml so 0.3 litres = 0.3 × 1000 = 300 ml of wormer in the bottle
52 goats at 21 kg each is 52 × 21 = 1092 kg total weight of goats
To find the number of 4 ml doses required do:
- 1092 ÷ 15 = 72.8
- 72.8 × 4 ml = 291.2 ml of wormer needed.
Yes, David has enough wormer.

Hopefully you will now be feeling confident with converting units of measure within the same system. You’ll need to know how to do this and be able to remember how to convert from one to another by multiplying or dividing. If you need further practice converting between units, please revisit the ‘Units of measure’ session in the Everyday Maths 1 course.

If you are being asked to convert between units in different systems (e.g. between litres and gallons) or between currency (e.g. US dollars and British pounds), you won’t be expected to know the conversion rate – you’ll be given it in the question. The next section will show you how to use these conversion rates and you’ll be able to practise solving problems that require you to do this.

Summary

In this section you have learned:

that different units are used for measurement
unit selection depends on the item or object being measured
how to convert between units in the same system.

2 Converting currencies and measures between different systems

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

When working with measures you’ll often need to be able to convert between units that are in different systems (kilograms to pounds for example) or currencies (British pounds to euros for example). Whilst this might sound tricky, it’s really just using your multiplication and division skills. Let’s begin by looking at converting currencies.

2.1 Converting currencies

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Often when you are abroad, or buying something from the internet from another country, the price will be given in a different currency. You’ll want to be able to work out how much the item costs in your own currency so that you can make a decision about whether to buy the item or not. When changing money for a holiday, you may also need to work out how much currency you will get for the amount of pounds sterling you are changing.

Figure _unit3.2.1 Figure 7 Different currency notes

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A pile of notes in different currencies.

Figure 7 Different currency notes

Let’s look at an example.

Case study _unit3.2.1 Example: Pounds and euros

To convert between pounds and euros, the first thing you need to know is the exchange rate. Exchange rates change daily. If you are given a currency conversion question, though, you will be given the exchange rate to use. In this example we will use the exchange rate of:

£1 = €1.15

So how do you convert?

Method

Look at the diagram below. In order to get from 1 to 1.15, you must multiply by 1.15. Following that, to get from 1.15 to 1, you must divide by 1.15.

In order to go from pounds to euros then, you multiply by 1.15 and to convert from euros to pounds, you divide by 1.15.

Figure _unit3.2.2 Figure 8 A conversion rate for pounds and euros

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A conversion rate for pounds and euros. £ x 1.15 for €; € ÷ 1.15 for £.

Figure 8 A conversion rate for pounds and euros

a.You decide you are going to change £300 into euros for your holiday. Using the exchange rate of £1 = €1.15, how many euros will you get?
Referring back to the diagram above, you want to convert pounds to euros in this example so you must multiply by 1.15.
However, you will not always have a diagram to refer to which means that you will need to be able to decide yourself whether you need to multiply by the conversion rate or divide by it. We will now look at the method to help you to do this.
Method
Write down your conversion rate with the single unit on the left-hand side:
- £1 = €1.15
Next, you write down the figure you are converting under the correct side. In this case it is the £300 cash you are changing, so this figure needs to go under the pounds (£) side:
- £1 = €1.15
- £300 = €?
This will help you to work out whether you need to go forwards (in which case you multiply by the conversion rate) or backwards (in which case you divide by the conversion rate).
Here, you are converting from pounds to euros. This means you are going forwards so you need to multiply by the conversion rate:
- £1 = €1.15
- £300 = €?
- → forwards →
Therefore, you must multiply by 1.15:
- £300 × 1.15 = €345
Let’s apply the method to another example.
b.You are in Spain and want to buy a souvenir. The price is €35 (35 euros). You know that £1 = €1.15.
Method
Write the currency conversion rate down first with the single unit of measure (the 1) on the left hand-side.
- £1 = €1.15
Next, write down the figure you are converting under the correct side. In this case it is the cost of the souvenir (€35), so this figure needs to go on the euros (€) side:
- £1 = €1.15
- £? = €35
Here, you are converting from euros to pounds. This means you are going backwards so you need to divide by the conversion rate:
- £? = €35
- ← backwards ← therefore divide
- €35 ÷ 1.15 = £30.43 (rounded to two decimal places)

Before you try some for yourself, let’s look at one more example. This time involving pounds and dollars.

Case study _unit3.2.2 Example: Pounds and dollars

a.You want to buy an item from America. The cost is $120. You know that £1 = $1.30. How much does the item cost in pounds?
Method
Similarly to before, you can see from the diagram that in order to go from dollars to pounds you must divide by 1.30.
Figure _unit3.2.3 Figure 9 A conversion rate for pounds and dollars
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A conversion rate for pounds and dollars. £ x 1.30 for $; $ ÷ 1.30 for £
Figure 9 A conversion rate for pounds and dollars
To work out $120 in pounds then, you do:
- $120 ÷ 1.30 = £92.31 (rounded to two decimal places)
To figure this out without the aid of a diagram, write down your conversion rate:
- £1 = $1.30
Now write down the figure you want to convert (in this case the $120) under the correct side:
- £1 = $1.30
- £? = $120
As you are converting dollars to pounds, you are going backwards so need to divide by 1.30:
- $120 ÷ 1.30 = £92.31 (rounded to two decimal places)
b.You are sending a relative in America £20 for their birthday. You need to send the money in dollars. How much should you send?
Method
To convert from pounds to dollars, you need to write down you conversion rate:
- £1 = $1.30
Now write down the figure you want to convert (in this case the £20) under the correct side:
- £1 = $1.30
- £20 = $?
As you are converting pounds to dollars, you are going forwards so need to multiply by 1.30:
- £20 × 1.30 = $26
You need to send $26.

Box _unit3.2.1 Summary of method

Write your conversion rate down first, making sure you put the single unit (the 1) on the left-hand side. This will make it easier to decide whether you need to multiply by the conversion rate or divide by it.

If you are going forwards, you multiply by the conversion rate.
If you are going backwards, you divide by the conversion rate.

Activity _unit3.2.1 Activity 4: Currency conversions

Calculate the answers to the following questions without using a calculator. You may double-check your answers on a calculator if needed.

Sarah sees a handbag for sale on a cruise ship for €80. She sees the same handbag online for £65. She wants to pay the cheapest possible price for the bag. On that day the exchange rate is £1 = €1.25. Where should she buy the bag from?
Josh is going to South Africa and wants to change £250 into rand. He knows that £1 = 18.7 rand. How many rand will he get for his money?
Alice is returning from America and has $90 left over to change back into £s. Given the exchange rate of £1 = $1.27, how many £s will Alice receive? Give your answer rounded to two decimal places.

Answer

For this question you can either convert from £ to € or from € to £.
Converting from £ to €:
- £1 = €1.25
- £65 = €?
As you are converting pounds to euros, you are going forwards so need to multiply by 1.25:
- £65 × 1.25 = €81.25
Sarah should buy the bag on the ship.
Converting from € to £:
- £1 = €1.25
- £? = €80
As you are converting euros to pounds you are going backwards so need to divide by 1.25:
- €80 ÷ 1.25 = £64
Sarah should buy the bag on the ship.
You need to convert from £ to rand so you do:
- £1 = 18.7 rand
- £250 = ? rand
As you are converting pounds to rand you are going forwards so need to multiply by 18.7:
- £250 × 18.7 = 4675 rand
You need to convert from $ to £ so you do:
- £1 = $1.27
- £? = $90
As you are converting dollars to pounds, you are going backwards so need to divide by 1.27:
- $90 ÷ 1.27 = £70.87 (rounded to two d.p.)

Hopefully you are now feeling confident with converting between currencies. The next part of this section deals with converting other units of measure between different systems – centimetres to inches, pounds to kilograms etc. These conversions are incredibly similar to converting currency and the conversion rate will always be given to you so it’s not something you have to remember.

2.2 Converting units of measure between different systems

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

This is a useful skill to have because you may often measure something in one unit, say your height in feet and inches, but then need to convert it to centimetres. You may, for example, be training to run a 10 km race and want to know how many miles that is. To do this all you need to know is the conversion rate and then you can use your multiplication and division skills to calculate the answer.

When performing these conversions, you are converting between metric and imperial measures. Metric measures are commonly used around the world but there are some countries (the USA for example) who still use imperial units. Whilst the UK uses metric units (g, kg, m, cm etc.) there are some instances where you may still need to convert a metric measure to an imperial one (inches, feet, gallons etc.).

In most cases, the conversion rate will be given in a similar format to the way money exchange rates are given, i.e. 1 inch = 2.5 cm or 1 kg = 2.2 lb. In these cases you can do exactly the same as you would with a currency conversion. The only exceptions here are where you don’t have one of the units of measure as a single unit i.e. 5 miles = 8 km.

As this skill is very similar to the one you previously learnt with currency conversions, let’s just look at two brief examples before you have a go at an activity for yourself.

Case study _unit3.2.3 Example: Centimetres and inches

You want to start your own business making accessories. One of the items you will be making is tote bags. The material you have bought is 156 cm wide. You need to cut pieces of material that are 15 inches wide.

How many pieces can you cut from the material you bought?

Use 1 inch = 2.54 cm.

Method

You need to start by converting either cm to inches or inches to cm so that you are working with units in the same system. Let’s look at both ways so that you feel confident that you will get the same answer either way.

Figure _unit3.2.4 Figure 10 Converting between inches and centimetres

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A conversion chart for converting inches to centimetres. inches x 2.54 for cm; cm ÷ 2.54 for inches.

Figure 10 Converting between inches and centimetres

You can see from the diagram above that to convert from inches to cm you must multiply by 2.54. To convert from cm to inches you must divide by 2.54.

As with currency conversions, if you do not have a diagram to help you, you will have to work out yourself whether you need to multiply by the conversion rate or divide by it. To do this, write down your conversion rate:

Extract _unit3.2.1

1 inch = 2.54 cm (remember to put the single unit – the 1 – on the left-hand side)

Converting from inches to cm

Now write the figure you are converting under the correct side:

Extract _unit3.2.2

1 inch = 2.54 cm

15 inches = ? cm

Since you want to know what 15 inches is in centimetres, you are going forwards so you need to multiply by the conversion rate of 2.54:

Extract _unit3.2.3

15 inches × 2.54 = 38.1 cm. This is the length of the material needed for each bag.

To calculate how many pieces of material you can cut you then do:

Extract _unit3.2.4

156 ÷ 38.1 = 4.0944…. As you need whole pieces of material, you need to round this answer down to 4 pieces.

Converting from cm to inches

You may have decided to convert from centimetres to inches instead. You still need to write down your conversion rate first:

Extract _unit3.2.5

1 inch = 2.54 cm (remember to put the single unit – the 1 – on the left-hand side)

Now write the figure you are converting under the correct side (in this case you will convert the 156 centimetres into inches):

Extract _unit3.2.6

1 inch = 2.54 cm

? inches = 156 cm

Since you want to know what 156 centimetres is in inches, you are going backwards so you need to divide by the conversion rate of 2.54:

Extract _unit3.2.7

156 ÷ 2.54 = 61.417…, this is the length of the big piece of material.

To calculate how many pieces of material you can cut you then do:

Extract _unit3.2.8

61.417… ÷ 15 = 4.0944…. Again, as you need whole pieces of material, you need to round this answer down to 4 pieces.

You can see that no matter which way you choose to convert you will arrive at the same answer.

Box _unit3.2.2

Note: Some numbers do not divide into each other exactly so you end up with lots of digits after the decimal point. Where this is the case, you need to round off the answer. You may be told what to round off to in the question (e.g. to the nearest whole number or to two decimal places) or you may have to use your own judgement.

For money you will normally round to two decimal places. In this case you are thinking about whole pieces of material – you only have four whole pieces so this is what you round to here.

Case study _unit3.2.4 Example: Kilometres and miles

You have signed up for a 60 km bike ride. There is a lake in a nearby park that you want to use for your training. You know that one lap of the lake is 2 miles. You want to cycle a distance of at least 40 km in your last training session before the race. How many full laps of the lake should you do?

You know that a rough conversion is 5 miles = 8 km.

There is more than one way to go about this calculation and, as ever, if you have a different method that works for you and you arrive at the same answer, feel free to use it!

Method

Since you already know how to convert if you are given a conversion that has a single unit of whichever measure you are using, it makes sense to just change the given conversion into one that you are used to dealing with.

Look at the diagram below – since you know that 5 miles is worth 8 km, you can find out what 1 mile is worth by simply dividing 5 by 5 so that you arrive at 1 mile. Whatever you do to one side, you must do to the other. Therefore, you also do 8 divided by 5. This then gives 1 mile = 1.6 km.

Figure _unit3.2.5 Figure 11 Calculating 1 mile in kilometres

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Diagram showing how to calculate 1 mile in kilometres using the information that 5 miles equals 8 kilometres. Arrows show that dividing both 5 miles by 5 and 8 kilometres by 5 gives ‘1 mile = 1.6 km’.

Figure 11 Calculating 1 mile in kilometres

Now that you know 1 mile = 1.6 km, you can solve this problem in its usual way:

Figure _unit3.2.6 Figure 12 Converting between miles and kilometres

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A conversion chart for miles and kilometres. miles x 1.6 for km; km ÷ 1.6 for miles.

Figure 12 Converting between miles and kilometres

Write down your conversion rate:

Extract _unit3.2.9

1 mile = 1.6 km

Now write down the figure you want to convert under the correct side (in this case you want to convert 40 km into miles):

Extract _unit3.2.10

1 mile = 1.6 km

? miles = 40 km

Since you first want to know what 40 km is in miles, you are going backwards, so you need to divide by 1.6:

Extract _unit3.2.11

40 ÷ 1.6 = 25 miles

So, 40 km = 25 miles

You know that one lap of the lake is 2 miles so:

Extract _unit3.2.12

25 ÷ 2 = 12.5 laps

Therefore, you will need to cycle 12.5 laps around the lake in order to have cycled your target of 40 km.

Now that you have seen a couple of worked examples, have a go at the activity below to check your understanding.

Activity _unit3.2.2 Activity 5: Converting between systems

Wherever possible, please do the calculations first without a calculator. You may then double-check on a calculator if needed.

A Ford Fiesta car can hold 42 litres of petrol. Using the fact that 1 gallon = 4.54 litres, work out how many gallons of petrol the car can hold. Give your answer rounded to two decimal places.
The café you work at has run out of milk and you have been asked to go to the shop and buy 10 pints. When you arrive however, the milk is only available in 2 litre bottles. You know that 1 litre = 1.75 pints. How many 2 litre bottles should you buy?
You are packing to go on holiday and are allowed 25 kg of luggage on the flight. You’ve weighed your case on the bathroom scales and it weighs 3 stone and 3 pounds.
You know that:
- 1 stone = 14 pounds
- 2.2 pounds = 1 kg
Is your luggage over the weight limit?
Your son wants to go on a ride at a theme park that has a minimum height restriction of 122 cm. You know your son is 4 ft. 7 in. tall. You know that:
- 1 foot (ft.) = 12 inches
- 1 inch (in.) = 2.54 cm
Can your child go on the ride?

Answer

You do:
- 1 gallon = 4.54 litres
- ? gallons = 42 litres
As you are converting litres to gallons, you are going backwards so need to divide by 4.54.
- 42 ÷ 4.54 = 9.25 gallons (to two d.p)
Firstly, you want to know how many litres there are in 10 pints, so you need to convert from pints to litres. You do:
- 1 litre = 1.75 pints
- ? litres = 10 pints
As you are converting pints to litres, you are going backwards so need to divide by 1.75:
- 10 ÷ 1.75 = 5.71… litres. As you can only buy the milk in 2 litre bottles, you will have to buy 6 litres of milk in total. You therefore need:
- 6 ÷ 2 = 3 bottles of milk
Firstly, you need to convert 3 stone and 3 pounds into just pounds. Since 1 stone = 14 pounds, to work out 3 stone you do:
- 3 × 14 = 42 pounds
Then you need to add on the extra 3 pounds, so in total you have 42 + 3 = 45 pounds.
Now that you know this, you can convert from pounds to kg.
Since 2.2 pounds = 1 kg, you can also say that 1 kg = 2.2 pounds (this doesn’t change anything, it just makes it a little easier as you can stick to your usual method of having the single unit on the left hand side):
- 1 kg = 2.2 pounds
- ? kg = 45 pounds
As you are converting from pounds to kg, you are going backwards so need to divide by 2.2:
- 45 ÷ 2.2 = 20.45 kg
Yes, your luggage is within the weight limit.
Firstly, you need to convert 4 ft. 7 in. into just feet. Since 1 foot = 12 inches, to work out 4 feet you do:
- 4 × 12 = 48 inches
Then you need to add on the extra 7 inches, so in total your son is 48 + 7 = 55 inches tall.
Now that you know this, you can convert from inches to centimetres:
- 1 inch = 2.54 cm
- 55 inches = ? cm
As you are converting from inches to centimetres, you are going forwards so need to multiply by 2.54:
- 55 × 2.54 = 139.7 cm
So your son is tall enough to go on the ride.

You should now be feeling confident with your conversion skills so it’s time to move on to the next part of this session.

Summary

In this section you have learned:

that different systems of measurement can be used to measure the same thing (e.g. a cake could be weighed in either grams or pounds)
you can convert between these systems using your multiplication and division skills and the given conversion rate
currencies can be converted in the same way as long as you know the exchange rate – this is particularly useful for holidays.

3 Time, timetables and average speed

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Calculating with time is often seen as tricky, not surprising really considering how difficult it can be to learn how to tell the time. The reason many people find calculating with time tricky is because, unlike nearly every other mathematical concept, it does not work in 10s. Time works in 60s – 60 seconds in a minute, 60 minutes in an hour. You cannot therefore, simply use your calculator to add on or subtract time.

Figure _unit3.3.1 Figure 13 A radio alarm clock

Think about this simple example. If it’s 9:50 and your bus takes 20 minutes to get to work, you cannot work out the time you will arrive by doing 950 + 20 on your calculator. This would give you an answer of 970 or 9:70 – there isn’t such a time!

You will need to calculate with time and use timetables in daily life to complete basic tasks such as: getting to work on time, working out which bus or train to catch, picking your children up from school on time, cooking and so many other daily tasks.

3.1 Calculating with time and timetables

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

As previously discussed, calculators are not the most useful items when it comes to calculations involving time. A much better option is to use a number line to work out these calculations. Take a look at the examples below.

Case study _unit3.3.1 Example: Cooking

You put a chicken in the oven at 4:45 pm. You know it needs to cook for 1 hour and 25 minutes. What time should you take the chicken out?

Method

Watch the video below to see how the number line method works.

Download this video clip.Video player: ani11_maths_chicken.mp4

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Transcript

A chicken takes 1 hour and 25 minutes to cook. So, to work out when the chicken will be ready, you can use a number line, which looks like this. The idea behind the number line is that we use small, easy chunks of time to work out the answer. In this example, you know that you need to add 1 hour and 25 minutes onto 4:45 p.m. Here is a number line that starts at 4:45 p.m.

To begin with, add 15 minutes, since this will take us to an easy time of 5:00 p.m. It then makes sense to add an hour on, which takes you to 6:00 p.m. As you've now added 1 hour and 15 minutes, you still need to add another 10 minutes. This takes you to 6:10 p.m., which is when the chicken will be ready. There's no exact science to using the number line for calculations like these. You just add on in chunks of time to make the calculation simpler.

End transcript

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Case study _unit3.3.2 Example: Time sheets

You work for a landscaping company and need to fill out your time sheet for your employer. You began working at 8:30 am and finished the job at 12:10 pm. How long did the job take?

Method

Figure _unit3.3.2 Figure 14 A number line for a time sheet

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A number line with times marked from left to right: 8:30am, 9:00am, 12:00pm, 12:10pm. An arrow labelled ‘+ 30 mins’ points from 8:30am, to 9:00am. An arrow labelled ‘+ 3 hours’ points from 9:00am to 12:00pm. An arrow labelled ‘+ 10 mins’ points from 12:00pm to 12:10pm.

Figure 14 A number line for a time sheet

Again, for finding the time difference you want to work with easy ‘chunks’ of time. Firstly, you can move from 8:30 am to 9:00 am by adding 30 minutes. It is then simple to get to 12:00 pm by adding on 3 hours.

Finally, you just need another 10 minutes to take you to 12:10 pm. Looking at the total time added you have 3 hours and 40 minutes.

Another aspect of calculating with time comes in the form of timetables. You will be used to using these to work out which departure time you need to meet in order to get to a location on time or how long a journey will take. Once you can calculate with time, using timetables simply requires you to find the correct information before carrying out the calculation. Take a look at the example below.

Case study _unit3.3.3 Example: Timetables

Here is part of a train timetable from Swindon to London.

Table _unit3.3.1 Table 2(a)
Swindon	06:10	06:27	06:41	06:58	07:01	07:17
Didcot	06:27	06:45	06:58	07:15	07:18	07:34
Reading	06:41	06:59	07:13	-	07:33	-
London	07:16	07:32	07:44	08:02	08:07	08:14

a.You need to travel from Didcot to London. You need to arrive in London by 8:00 am. What is the latest train you can catch from Didcot to arrive in London for 8:00 am?

Method

Table _unit3.3.2 Table 2(b)
Swindon	06:10	06:27	06:41	06:58	07:01	07:17
Didcot	06:27	06:45	06:58	07:15	07:18	07:34
Reading	06:41	06:59	07:13	-	07:33	-
London	07:16	07:32	07:44	08:02	08:07	08:14

Looking at the arrival times in London, in order to get there for 8:00 am you will need to take the train that arrives in London at 07:44 (highlighted with bold). If you then move up this column of the timetable you can see that this train leaves Didcot at 06:58 (highlighted with italic). This is therefore the train you must catch.

b.How long does the 06:58 from Swindon take to travel to London?

Method

Table _unit3.3.3 Table 2(c)
Swindon	06:10	06:27	06:41	06:58	07:01	07:17
Didcot	06:27	06:45	06:58	07:15	07:18	07:34
Reading	06:41	06:59	07:13	-	07:33	-
London	07:16	07:32	07:44	08:02	08:07	08:14

Firstly, find the correct train from Swindon (highlighted with italic). Follow this column of the timetable down until you reach London (highlighted with bold). You then need to find the difference in time between 06:58 and 08:02. Using the number line method from earlier in the section (or any other method you choose).

Figure _unit3.3.3 Figure 15 A number line for a timetable

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A number line with times marked from left to right: 06:58, 07:00, 08:00, 08:02. An arrow labelled ‘+ 2 mins’ points from 06:58, to 07:00. An arrow labelled ‘+ 1 hour’ points from 07:00 to 08:00. An arrow labelled ‘+ 2 mins’ points from 08:00 to 08:02.

Figure 15 A number line for a timetable

You can then see that this train takes a total of 1 hour and 4 minutes to travel from Swindon to London.

Have a go at the activity below to practise calculating time and using timetables.

Activity _unit3.3.1 Activity 6: Timetables and calculating time

Kacper is a builder. He leaves home at 8:30 am and drives to the trade centre. He collects his items and loads them into his van. His visit takes 1 hour and 45 minutes. He then drives to work, which takes 50 minutes. What time does he arrive at work?
You have invited some friends round for dinner and find a recipe for roast lamb. The recipe requires:
- 25 minutes preparation time
- 1 hour cooking time
- 20 minutes resting time
You want to eat with your friends at 7:30 pm. What is the latest time you can start preparing the lamb?
Here is part of a train timetable from Manchester to Liverpool.

Table _unit3.3.4 Table 3(a)
Manchester to Liverpool
Manchester	10:24	10:52	11:03	11:25	12:01	12:13
Warrington	10:38	11:06	11:20	11:45	12:15	12:28
Widnes	10:58	11:26	11:42	12:03	12:34	12:49
Liverpool Lime Street	11:09	11:38	11:53	12:14	12:46	13:02

You need to travel from Manchester to Liverpool Lime Street. You need to be in Liverpool by 12:30. Which train should you catch from Manchester and how long will your journey take?

Answer

Firstly, work out the total time that Kacper is out for:
- 1 hour 45 minutes at the trade centre and another 50 minutes driving makes a total of 2 hours and 35 minutes.
Then, using the number line, you have:
Figure _unit3.3.4 Figure 16 A number line for Question 1
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A number line with times marked from left to right: 8:30am, 9:00am, 10:00am, 11:00am, 11:05am. An arrow labelled ‘+ 30 mins’ points from 8:30am, to 9:00am. An arrow labelled ‘+ 1 hour’ points from 9:00am to 10:00am. An arrow labelled ‘+ 1 hour’ points from 10:00am to 11:00am An arrow labelled ‘+ 5 mins’ points from 11:00am to 11:05am.
Figure 16 A number line for Question 1
So Kacper arrives at work at 11:05 am.
You could also do the calculation by adding on the 1 hour 45 minutes first:
- 8:30 am + 1 hour = 9:30 am
- 9:30 am + 45 minutes = 10:15 am
Finally, you can add on the 50 minutes:
- 10:15 am + 45 minutes = 11:00 am
Then add on the remaining 5 minutes:
- 11:00 am + 5 minutes = 11:05 am
Again, firstly work out the total time required:
- 25 minutes + 1 hour + 20 minutes = 1 hour 45 minutes in total
This time you need to work backwards on the number line so you begin at 7:30 and work backwards.
Figure _unit3.3.5 Figure 17 A number line for Question 2
Show description|Hide description
A number line with times marked from left to right: 5:45pm, 6:00pm, 7:00pm, 7:30pm. An arrow labelled ‘– 30 mins’ points from 7:30pm, to 7:00pm. An arrow labelled ‘‒ 1 hour’ points from 7:00pm to 6:00pm. An arrow labelled ‘‒ 15 mins’ points from 6:00pm to 5:45pm.
Figure 17 A number line for Question 2
You can now see that you must begin preparing the lamb at 5:45 pm at the latest.
As with the first question, you could have done this question by taking off each stage in the cooking process separately rather than finding the total time first:
- 7:30 pm − 20 minutes = 7:10 pm
- 7:10 pm − 1 hour = 6:10 pm
There are 25 minutes left so:
- 6:10 pm − 10 minutes = 6:00 pm
There are now 15 minutes left so:
- 6:00 pm − 15 minutes = 5:45 pm
Looking at the timetable for arrival at Liverpool, you can see that in order to arrive by 12:30 you need to catch the train that arrives at 12:14. This means that you need to catch the 11:25 from Manchester.

Table _unit3.3.5 Table 3(b)
Manchester to Liverpool
Manchester	10:24	10:52	11:03	11:25	12:01	12:13
Warrington	10:38	11:06	11:20	11:45	12:15	12:28
Widnes	10:58	11:26	11:42	12:03	12:34	12:49
Liverpool Lime Street	11:09	11:38	11:53	12:14	12:46	13:02

You therefore need to work out the difference in time between 11:25 (italic) and 12:14 (bold).

Figure _unit3.3.6 Figure 18 A number line for Question 3

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A number line with times marked from left to right: 11:25, 11:30, 12:00, 12:14. An arrow labelled ‘+ 5 mins’ points from 11:25, to 11:30. An arrow labelled ‘+ 30 mins’ points from 11:30 to 12:00. An arrow labelled ‘+ 14 mins’ points from 12:00 to 12:14.

Figure 18 A number line for Question 3

Using the number line again, you can see that this is a total of 5 + 30 + 14 = 49 minutes.

You should now be feeling comfortable with calculations involving time and timetables. Before you move on to looking at problems that involve average speed, it is worth taking a brief look at time conversions. Since you are already confident with converting units of measure, this part will just consist of a brief activity so that you can practise converting units of time.

3.2 Converting units of time

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

You can see from the diagram below that to convert units of time you can use a very similar method to the one you used when converting other units of measure. There is one slight difference when working with time however.

Figure _unit3.3.7 Figure 19 A conversion chart for time

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A conversion chart for time. weeks × 7 for days; days ÷ 7 for weeks days × 24 for hours; hours ÷ 24 for days hours × 60 for minutes; minutes ÷ 60 for hours minutes × 60 for seconds; seconds ÷ 60 for minutes.

Figure 19 A conversion chart for time

Let’s say you want to work out how long 245 minutes is in hours. The diagram above shows that you should do 245 ÷ 60 = 4.083. This is not a particularly helpful answer since you really want the answer in the format of: ___ hours __ minutes. Due to the fact that time does not work in 10s, you need to do a little more work once arriving at your answer of 4.083.

The answer is obviously 4 hours and an amount of minutes.

4 hours then is 4 × 60 = 240 minutes.

Since you wanted to know how long 245 minutes is you just do 245 – 240 = 5 minutes left over. So 245 minutes is 4 hours and 5 minutes.

It’s a very similar process if you want to go from say minutes to seconds. Let’s take it you want to know how long 5 minutes and 17 seconds is in seconds. 5 minutes would be 5 × 60 = 300 seconds. You then have a further 17 seconds to add on so you do 300 + 17 = 317 seconds.

Have a go at the activity below to make sure you feel confident with converting times.

Activity _unit3.3.2 Activity 7: Converting times

Convert the following times:

6 hours and 35 minutes = ___ minutes.
85 minutes = ____ hours and ____ minutes.
153 seconds = ____ minutes and ___ seconds.
46 days = ___ weeks and ____days.
3 minutes and 40 seconds = ____ seconds.

Answer

6 hours = 6 × 60 = 360 minutes
360 minutes + 35 minutes = 395 minutes
85 minutes ÷ 60 = 1.417 (rounded to three d.p)
1 hour = 60 minutes.
85 minutes − 60 minutes = 25 minutes remaining
So 85 minutes = 1 hour and 25 minutes
153 seconds ÷ 60 = 2.55
2 minutes = 2 × 60 = 120 seconds
153 seconds − 120 seconds = 33 seconds remaining
So 153 seconds = 2 minutes and 33 seconds
46 days ÷ 7 = 6.571 (rounded to three d.p)
6 weeks = 6 × 7 = 42 days
46 days − 42 days = 4 days remaining
So 46 days = 6 weeks and 4 days
3 minutes = 3 × 60 = 180 seconds
180 seconds + 40 seconds = 220 seconds

Hopefully you found that activity fairly straightforward and are now feeling ready to move on to the next part of the ‘Units of measure’ session – Average speed.

3.3 Average speed

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

The sign below is commonly seen on motorways but it is not the only time when it is useful to know your average speed.

Figure _unit3.3.8 Figure 20 A speed camera sign

Being able to calculate and use average speed can help you to work out how long a journey is likely to take. The method for working out average speed involves using a simple formula.

Figure _unit3.3.9 Figure 21 A formula for average speed

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Illustration of a car driving for a distance past a speed camera. The text shows the formula ‘Speed equals distance divided by time’.

Figure 21 A formula for average speed

You can also use this formula to work out the distance travelled when given a time and the average speed, or the time taken for a journey when given the distance and average speed.

The formulas for this are shown in the diagram below. You can see that when given any two of the elements from distance, speed and time, you will be able to work out the third.

Figure _unit3.3.10 Figure 22 Distance, speed and time formulas

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Illustration of 3 triangles that show the relationship between distance, speed and time. The text reads: ‘Distance equals Speed multiplied by Time’ ‘Time equals Distance divided by Speed’ ‘Speed equals Distance divided by Time’.

Figure 22 Distance, speed and time formulas

If you can learn this formula triangle, when you want to use it, you write it down and cover up what you want to work out (the segment in orange). This will tell you what calculation you need to do.

Let’s look at an example of each so that you can familiarise yourself with it.

Case study _unit3.3.4 Example: Calculating distance

A car has travelled at an average speed of 52 mph over a journey that lasts 2 and a half hours. What is the total distance travelled?

Method

You can see that to work out the distance you need to do speed × time. In this example then we need to do 52 × 2.5. It is very important to note here that 2 and a half hours must be written as 2.5 (since 0.5 is the decimal equivalent of a half).

You cannot write 2.30 (for 2 hours and 30 mins). If you struggle to work out the decimal part of the number, convert the time into minutes (2 and a half hours = 150 minutes) and then divide by 60 (150 ÷ 60 = 2.5).

Extract _unit3.3.1

52 × 2.5 = 130 miles travelled

Case study _unit3.3.5 Example: Calculating time

A train will travel a distance of 288 miles at an average speed of 64 mph. How long will it take to complete the journey?

Method

You can see from the formula that to calculate time you need to do distance ÷ speed so you do:

Extract _unit3.3.2

288 ÷ 64 = 4.5 hours

Again, note that this is not 4 hours 50 minutes but 4 and a half hours.

If you are unsure of how to convert the decimal part of your answer, simply multiply the answer by 60, which will turn it into minutes and you can then convert from there.

In this case, 4.5 × 60 = 270 minutes. We already know from the answer of 4.5 hours that this is 4 whole hours and so many minutes, so we now need to work out how many minutes the .5 represents:

Extract _unit3.3.3

60 × 4 = 240 minutes

270 − 240 = 30 minutes

So 4.5 hours = 270 minutes = 4 hours, 30 minutes

Case study _unit3.3.6 Example: Calculating speed

A Formula One car covers a distance of 305 km during a race. The time taken to finish the race is 1 hour and 15 minutes. What is the car’s average speed?

Method

The formula tells you that to calculate speed you must do distance ÷ time. Therefore, you do 305 ÷ 1.25 (since 15 minutes is a quarter of an hour and 0.25 is the decimal equivalent of a quarter):

Extract _unit3.3.4

305 ÷ 1.25 = 244 km/h

In a similar way to example 1, if you are unsure of how to work out the decimal part of the time simply write the time (in this case 1 hour and 15 minutes) in minutes, (1 hour 15 minutes = 75 minutes) and then divide by 60:

Extract _unit3.3.5

75 ÷ 60 = 1.25

Now have a go at the following activity to check that you feel confident with finding speed, distance and time. Please do the calculations first without a calculator. You may then double-check on a calculator if needed.

Activity _unit3.3.3 Activity 8: Calculating speed, distance and time

Filip is driving a bus along a motorway. The speed limit is 70 mph. In 30 minutes, he travels a distance of 36 miles. Does his average speed exceed the speed limit?
A plane flies from Frankfurt to Hong Kong. The flight time was 10 hours and 45 minutes. The average speed was 185 km/h. What is the distance flown by the plane?
Malio needs to get to a meeting by 11:00 am. The time now is 9:45 am. The distance to the meeting is 50 miles and he will be travelling at an average speed of 37.5 mph. Will he be on time for the meeting?

Answer

You need to find the speed so you do: distance ÷ time.
The distance is 36 miles. The time is 30 minutes but you need the time in hours:
- 30 minutes ÷ 60 = 0.5 hours
Now you do:
- 36 ÷ 0.5 = 72 mph
Yes, Filip’s average speed did exceed the speed limit.
You need to find the distance so you do:
- speed × time
- 10 hours 45 minutes = 10.75 hours
If you are unsure how to express this in hours, convert 10 hours 45 minutes all into minutes:
- 10 × 60 = 600 + 45 = 645 minutes
Then divide by 60:
- 645 ÷ 60 = 10.75 hours
Now to work out the distance do:
- speed × time = 185 × 10.75 = 1988.75 km from Frankfurt to Hong Kong
You need to find the time so you do:
- distance ÷ speed
- 50 ÷ 37.5 = 1.33 hours (rounded to two d.p)
- Note: The actual answer is 1.3333333 (the 3 is recurring or never-ending).
To convert this to minutes do:
- 1.33 × 60 = 79.8 minutes
- round 79.8 minutes up to 80 minutes
- 80 minutes = 1 hour and 20 minutes
If the time now is 9:45 am and his meeting is at 11:00 am, then it is only 1 hour, 15 minutes until his meeting, so no, Malio will not make the meeting on time.

Hopefully you will now be feeling more confident with calculations involving speed, distance and time. You will now move on to temperature conversions.

Summary

In this section you have learned how to:

use timetables to plan a journey and how to calculate time efficiently
convert between units of time by using multiplication and division skills
use the formula for calculating distance, speed and time.

4 Temperature

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Temperature can be recorded in either degrees Celsius (°C) or degrees Fahrenheit (°F). In Everyday Maths 1 you used conversion tables to help you to compare temperatures expressed in the different units. You will now look at how to convert between them using formulas.

4.1 Celsius and Fahrenheit formulas

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

The following formulas can be used to convert between Celsius and Fahrenheit.

Box _unit3.4.1

To convert Celsius to Fahrenheit use the formula:

F = C + 32
Method:
- divide the Celsius figure by 5
- multiply by 9
- add 32.
If you prefer, you can multiply the Celsius figure by 9 first and then divide by 5. You will still need to add on 32 at the end.

Box _unit3.4.2

To convert Fahrenheit to Celsius use the formula:

C =
Method:
- subtract 32 from the Fahrenheit figure
- multiply by 5
- divide by 9.

If you need a recap on the rules for using formulas, revisit Session 1 ‘Working with numbers’. We will now look at an example.

Case study _unit3.4.1 Example: Which city is warmer?

I look up the average temperature for New York on a particular day and it is 10°C. I know the average temperature in Swansea on the same day is 55°F. Which city is warmer?

You either need to convert New York’s temperature into °F or the Swansea temperature into °C.

Method 1 – Converting °C to °F

If we look back at the formulas above, the one we need to use to convert from °C to °F is:

Extract _unit3.4.1

F = C + 32

We need to substitute the C with our °C figure of 10°C. We then need to follow the rules of BIDMAS to carry out the calculation in stages, as shown below:

Extract _unit3.4.2

F = × 10 + 32

Divide the celsius figure by 5:

10 ÷ 5 = 2

Multiply by 9:

2 × 9 = 18

Add 32:

18 + 32 = 50°F

You may have done the calculation slightly differently by multiplying the Celsius figure by 9 first and then dividing by 5. The answer will work out the same:

Extract _unit3.4.3

F = × 10 + 32

Multiply by the Celsius figure by 9:

10 × 9 = 90

Divide by 5:

90 ÷ 5 = 18

Add 32:

18 + 32 = 50°F

So which is warmer:

New York at 10°C (which we now know is 50°F) or Swansea at 55°F?
Swansea is warmer.

Method 2 – Converting °F to °C

The formula for converting from °F to °C:

Extract _unit3.4.4

C =

We need to substitute the F with our °F figure of 55°F. We then need to follow the rules of BIDMAS to carry out the calculation in stages, as shown below:

Extract _unit3.4.5

Take 32 away from the Fahrenheit figure of 55:

55 − 32 = 23

Multiply by 5:

23 × 5 = 115

Divide by 9:

115 ÷ 9 = 12.8°C (rounded to 1 decimal place)

So which is warmer:

New York at 10°C or Swansea at 55°F (which we now know is 12.8°C)? Swansea is warmer.

Box _unit3.4.3

Hint: Google has its own unit converter (search for Google Unit Converter) which you can use to convert between various units of measure, including between °C and °F. You could try using it to double-check your answers to the questions below.

Activity _unit3.4.1 Activity 9: Temperature conversions

Work out the answers to the following without using a calculator. You may double-check your answers on a calculator or using the Google unit converter, if needed, and remember to check your answers with ours at the end. Round your answers off to one decimal place where needed.

Convert the following temperatures into degrees Fahrenheit (°F):
- a.22°C
- b.0°C
- c.−6°C

Answer

You need to use the following formula:
F = C + 32
- a.F = × 22 + 32
  22 ÷ 5 = 4.4
  4.4 × 9 = 39.6
  39.6 + 32 = 71.6°F
- b.F = × 0 + 32
  0 ÷ 5 = 0
  0 × 9 = 0
  0 + 32 = 32°F
- c.F = × −6 + 32
  −6 ÷ 5 = −1.2
  −1.2 × 9 = −10.8
  −10.8 + 32 = 21.2°F

Convert the following temperatures into degrees Celsius (°C):
- a.45°F
- b.212°F
- c.5°F

Answer

You need to use the following formula:
C =
- a.C =
  45 − 32 = 13
  13 × 5 = 65
  65 ÷ 9 = 7.2°C (to one d.p)
- b.C =
  212 − 32 = 180
  180 × 5 = 900
  900 ÷ 9 = 100°C
- c.C =
  5 − 32 = −27
  −27 × 5 = −135
  −135 ÷ 9 = −15°C

I find a recipe which states that my oven needs to be set at a temperature of 400°F. My settings on my oven are in °C. What temperature should I set my oven to?

Answer

You need to convert 400°F to °C so use the formula:
C =

C =

400 − 32 = 368
368 × 5 = 1840
1840 ÷ 9 = 204.4°C (to one d.p).
As you would be unable to set an oven so accurately, you would set the temperature to 200°C.

I see Moscow’s temperature is −4°C on a particular day in February, whilst the temperature in Toronto is 19°F. Which place is colder?

Answer

You either need to convert the Moscow temperature of −4°C to °F, or convert the Toronto temperature of 19°F to °C.
Method 1 – Converting °C to °F
If we look back at the formulas, the one we need to use to convert from °C to °F is:
- F = C + 32
We need to substitute the C with our °C figure of −4°C. We then need to follow the rules of BIDMAS to carry out the calculation in stages, as shown below:
- F = × −4 + 32
- −4 ÷ 5 = −0.8
Multiply by 9:
- −0.8 × 9 = −7.2
Add 32:
- −7.2 + 32 = 24.8°F
So which is colder? Moscow at −4°C (which we now know is 24.8°F) or Toronto at 19°F? Toronto is colder.
Method 2 – Converting °F to °C
The formula for converting from °F to °C is:
- C =
We need to substitute the F with our °F figure of 19°F. We then need to follow the rules of BIDMAS to carry out the calculation in stages, as shown below:
- C =
Take 32 away from the Fahrenheit figure of 19:
- 19 − 32 = −13
Multiply by 5:
- −13 × 5 = −65
Divide by 9:
- −65 ÷ 9 = −7.2°C (to one d.p.)
So which is colder: Moscow at −4°C or Toronto at 19°F (which we now know is −7.2°C)? Toronto is colder.

Hopefully you will be feeling more confident when solving problems relating to temperature. The next section will cover reading measurements on scales.

Summary

In this section you have:

practised converting between degrees Celsius (°C) and degrees Fahrenheit (°F).

5 Reading scales

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

You may need to read a scale to measure out an amount of liquid, read a measurement on a ruler, weigh out ingredients for a recipe or to take someone’s temperature.

Reading scales can be tricky because every scale is different.

To read a scale correctly, you need to ask yourself:

What does the scale go up in? What steps or intervals does it use?

Box _unit3.5.1

Note: The marks on a scale may be referred to as any of the following: intervals, steps, increments or markers. These terms are often used interchangeably.

5.1 Scale examples

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Take a look at the following examples.

Case study _unit3.5.1 Example 1: Reading scales

Figure _unit3.5.1 Figure 23 Scale with numbered intervals of 50

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There are four markers between each numbered interval on the scale. There are five numbered intervals going from 0–200. ‘(a)’ has an arrow that points to the second marker after ‘50’. ‘(b)’ has an arrow that points in between the first and second marker after ‘150’.

Figure 23 Scale with numbered intervals of 50

You can see that this scale is marked up in numbered intervals of 50. However, what does each line in between each numbered interval represent? You can use your judgement to help you to figure out what each small step represents.

Watch this video (https://corbettmaths.com/ 2013/ 04/ 27/ reading-scales/) for further information about how to do this.

Alternatively, you can work this out using division. If you count on from 0 to 50 on this scale, there are 5 steps: 50 ÷ 5 = 10, so each step is 10.

a.The arrow is pointing to the 2nd mark after 50. As the steps are going up in 10s, the arrow is pointing to 70.
b.The arrow is halfway between the 1st and 2nd step after 150. The first step is 160 and the second is 170 so the arrow is pointing to 165.

Case study _unit3.5.2 Example 2: Reading scales

Sometimes you will need to read scales where the reading will be a decimal number.

Figure _unit3.5.2 Figure 24 Scale with single numbered intervals going from 3–5

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There are nine markers between each numbered interval on the scale and the numbers 1–10 are displayed in red under the scale and count each marker between ‘3’ and ‘4’. There are three numbered intervals going from 3–5. ‘(a)’ has an arrow that points to the fourth marker after ‘3’. ‘(b)’ has an arrow that points to the eighth marker after ‘4’.

Figure 24 Scale with single numbered intervals going from 3–5

If you look at this scale, it goes up in numbered intervals of 1. From one whole number to the next whole number there are 10 small steps. 1 ÷ 10 = 0.1, so each step is 0.1.

Hint: Have a look at the image to show how to count the number of steps between the numbered markers.

a.The arrow is pointing to the fourth step after 3 so the arrow is pointing to 3.4.
b.The arrow is pointing to the eighth step after 4, so the arrow is pointing to 4.8.

5.2 Scales and measuring instruments

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Now you’ve worked through some examples have a go at the following activities.

Activity _unit3.5.1 Activity 10: Reading scales

Read the scales below and find the values the arrows are pointing to for (a), (b) and (c).
Figure _unit3.5.3 Figure 25 Scale with numbered intervals every hundred going from 100–400
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There are four numbered interval on the scale going from 100–400. There are four markers between each numbered interval. ‘(a)’ has an arrow that points to the third marker after ‘100’. ‘(b)’ has an arrow that points between the second and third marker after ‘200’. ‘(c)’ has an arrow that points to the fourth marker after ‘300’.
Figure 25 Scale with numbered intervals every hundred going from 100–400

Answer

The scale is going up in steps of 20 (the numbered markers are going up in intervals of 100 and there are 5 small steps between each numbered marker: 100 ÷ 5 = 20) so the answers are:
- a.160
- b.This is halfway between 240 and 260 so the answer is 250
- c.380

Read the values for (a), (b) and (c) on the scales below.
Figure _unit3.5.4 Figure 26 Scale with numbered intervals every hundred going from 1200–1500
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There are four numbered interval on the scale going from 1200–1500. There are three markers between each numbered interval. ‘(a)’ has an arrow that points to the second marker after ‘1200’. ‘(b)’ has an arrow that points to the first marker after ‘1300’. ‘(c)’ has an arrow that points to the third marker after ‘1500’.
Figure 26 Scale with numbered intervals every hundred going from 1200–1500

Answer

- a.1250
- b.1325
- c.1475

Read the values for (a), (b), (c) and (d) on the scales below.
Figure _unit3.5.5 Figure 27 Scale with single number intervals from 0–3
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There are four numbered interval on the scale going from 0–3. There are nine markers between each numbered interval. ‘(a)’ has an arrow that points to the fifth marker after ‘0’. ‘(b)’ has an arrow that points to the sixth marker after ‘1’. ‘(c)’ has an arrow that points to the third marker after ‘2’. ‘(d)’ has and arrow that points between the sixth and seventh marker after ‘2’.
Figure 27 Scale with single number intervals from 0–3

Answer

- a.0.5
- b.1.6
- c.2.3
- d.The arrow is pointing to halfway between 2.6 and 2.7 so the reading is 2.65.

Read the values for (a), (b), and (c) on the scales below.
Figure _unit3.5.6 Figure 28 Scale with numbered intervals every 5 going from 0–15
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There are four numbered intervals on the scale going from 0–15. There are nine markers between each numbered interval. ‘(a)’ has an arrow that points to the second marker after ‘0’. ‘(b)’ has an arrow that points to the fifth marker after ‘5’. ‘(c)’ has an arrow that points to the third marker after ‘10’.
Figure 28 Scale with numbered intervals every 5 going from 0–15

Answer

- a.1.0 (or just 1)
- b.7.5
- c.11.5

Now have a go at reading the scales on the different instruments of measure.

Activity _unit3.5.2 Activity 11: Measuring instruments

How much water is left in the bottle, to the nearest 50 millilitres (ml)?
Figure _unit3.5.7 Figure 29 A water bottle with a scale on the side and water inside
Show description|Hide description
There are twenty steps on the bottle and ‘1 litre’ on the top marker. There are ten larger, darker markers and a smaller marker in between each large one. The water line is very close to the fifth marker which is a smaller marker.
Figure 29 A water bottle with a scale on the side and water inside

Answer

The bottle holds 1 litre of liquid in total. There are 10 large steps marked on the bottle so each one marks 100 ml (1 litre = 1000 ml and 1000 ÷ 10 = 100).
Halfway between each large step there is a small step so each of these marks off 50 ml.
This means there is 250 ml of water left in the bottle to the nearest 50 ml.

Sara weighs her case using a set of luggage scales. She has a weight limit of 21 kg. How much more can she pack to the nearest 100 grams?
Figure _unit3.5.8 Figure 30 Luggage scales weighing luggage
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The illustration shows a hand holding analogue scales weighing some luggage. There are seven numbered intervals every five on the scales going from 0–30 kg with small markers every 1 kg. A square zooms in on the scales where nine smaller markers are shown between each 1 kg marker. In the zoomed square is a needle that points two smaller markers prior to the numbered interval of 20 kg.
Figure 30 Luggage scales weighing luggage

Answer

To answer this question you need to remember that 1 kg = 1000 g.
The scale is numbered at every 1 kg interval and there are 10 steps between each numbered interval, so each step marks 0.1 kg (1 ÷ 10 = 0.1). You could also think of each marker being 100 g (0.1 kg = 100 g).
The arrow is almost at 19.8 kg (19 800 g). If Sara has a weight limit of 21 kg then:
21 kg − 19.8 kg = 1.2 kg (21 000 g − 19 800 g = 1200 g)
Sara can pack another 1.2 kg (or 1200 g) worth of luggage.

Simon needs to weigh out 4 kg of potatoes. Looking at the reading on the scale, how many more grams of potatoes does he need to add to make 4 kg?
Figure _unit3.5.9 Figure 31 Food scales weighing potatoes
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Analogue scales are weighing potatoes. A zoomed in version of the scales reading is on the left of the image. There are six numbered intervals, one every kilogram on the scales going from 0–5 kg. There are nine unnumbered markers between each number interval. There is a needle that points two smaller markers prior to the numbered interval of 4 kg.
Figure 31 Food scales weighing potatoes

Answer

As with Question 2, you need to remember that 1 kg = 1000 g.
The scale is numbered at every 1 kg interval and there are 10 steps between each numbered marker so each step marks 0.1 kg (1 ÷ 10 = 0.1). You could also think of each step being 100 g (0.1 kg = 100 g).
The arrow is pointing to 3.8 kg (or 3800 g).
If Simon needs 4 kg (4000 g) of potatoes then he needs to weigh out another 200 g.

Hopefully you will be feeling confident at reading scales on measuring devices now which leads you nicely onto the next section which looks at conversion scales.

5.3 Using conversion scales

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Earlier on in the session you looked at converting between units of measure in different systems by carrying out calculations.

Many measuring instruments (e.g. thermometers, rulers, measuring jugs) have scales which show two or more different units of measure. This means that there may be times where you can compare the scales on the measuring instrument to make a conversion rather than carry out a calculation.

Look at the following example.

Figure _unit3.5.10 Figure 32 Example – Reading a thermometer

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A thermometer with ‘°C’ on the top-left and numbered intervals every ten going from ‘−30°’, displayed at the bottom to ‘50°’ at the top. ‘°F’ is on the top-right of the thermometer with numbered intervals every twenty going from ‘20°’displayed at the bottom to ‘120°’ at the top.

Figure 32 Example – Reading a thermometer

The thermometer above has a scale down the left-hand side which shows degrees Celsius (°C) and a scale on the right-hand side which shows degrees Fahrenheit (°F). This means that you can take a reading on this thermometer in both units of measure, depending on which is needed or which you are more familiar with. It can also help you to look up conversions between units.

You need to be careful with each scale, though – as they are showing different units, they are marked differently and go up in different steps.

On this thermometer, the degrees Celsius scale is going up in steps of 1°C, so the temperature shown is 38°C. If you want to take the reading in degrees Fahrenheit, you can see that it is 100°F (the scale is going up in steps of 2°F). It can be difficult to get a precise comparison between units, but using this thermometer, we can say that 38°C is roughly 100°F.

Now have a go at the following activity.

Activity _unit3.5.3 Activity 12: Using conversion scales

Look at the weighing scales below and answer the questions that follow.

View larger image

Figure _unit3.5.11 Figure 33 Weighing scales showing two units of measure

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A circular analogue weighing scales dial with ‘g’ on the outer circle of the dial in the 12 o’clock position on the dial. ‘oz’ is on the inner circle of the dial at the 12 o’clock position. On the ‘g’ scale (outer) there are numbered interval every fifty going from ‘0g’ to ‘450’. There are nine markers between each numbered interval. On the ‘oz’ (inner) scale there are single numbered intervals going from ‘0 oz’ to ‘17’ at the top with three markers between each numbered interval. The needle points to the fouth marker after ‘50’ on the ‘g’ scale and the second marker after ‘2’ on the ‘oz’ scale.

Figure 33 Weighing scales showing two units of measure

What is the reading shown by the arrow in grams?
How many ounces (oz) is 200 g, to the nearest whole oz?
Roughly how many grams is 1 oz, to the nearest 10 g?
I see a recipe which states that I need 6 oz of flour. Roughly, how many grams of flour is this?

Answer

Grams (g) are shown on the outside of this scale and ounces (oz) are shown on the inside.

The arrow is pointing to 70 g (the scale is going up in steps of 5).
You need to look on the outside of the scale to find 200 g and then look on the inside to see how many whole ounces it is nearest to.
The nearest whole ounce is 7 oz.
Find 1 oz on the inside of the scale. Now look on the outside to take this reading in grams. The nearest marker is 30 grams (the grams scale goes up in steps of 5) so 1 oz is approximately 30 g.
Look on the inside of the scale for 6 oz. Then take the equivalent gram reading from the outside of the scale. 6 oz is approximately 170 g.

You have now learned all you need to know about units of measures! If you feel unsure on any part of this section, feel free to refer back to the examples or activities again to ensure you feel secure in all areas. All that remains of this section is the end of session quiz. Good luck!

Summary

In this section you have learned to read:

measuring scales using different intervals
scales on different measuring instruments
conversion scales.

6 Session 2 quiz

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Now it’s time to review your learning in the end-of-session quiz.

Session 2 quiz.

Open the quiz in a new window or tab (by holding ctrl [or cmd on a Mac] when you click the link), then return here when you have done it.

7 Session 2 summary

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

You have now completed Session 2, ‘Units of measure’. If you have identified any areas that you need to work on, please ensure you refer to this section of the course and retry the activities.

You should now be able to:

understand that there are different units used for measuring and how to choose the appropriate unit
convert between measurements in the same system (e.g. grams and kilograms) and those in different systems (e.g. litres and gallons)
use exchange rates to convert currencies
work with time and timetables
work out the average speed of a journey using a formula
convert temperature measurements between Celsius (°C) and Fahrenheit (°F)
read scales on measuring equipment.

All of the skills listed above will help you with tasks in everyday life, such as measuring for new furniture or redesigning a room or garden. These are essential skills that will help you progress through your employment and education.

You are now ready to move on to Session 3, ‘Shape and space’.

Introduction

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Take a look at the picture below. In order to decorate the room, you would need to know the length of skirting required (perimeter), how much carpet to order and how many tins of paint to buy (area). You would also need to use your rounding skills (as items such as paint and skirting must be bought in full units) and your addition and multiplication knowledge – to work out the cost!

Figure 1 Floor plan of a house

This session of the course will draw upon the skills you learned in the ‘Working with numbers’ and ‘Units of measure’ sessions.

Throughout this session you will learn how to find the perimeter, area and volume of simple and more complex shapes – if you’ve ever decorated a room you will be familiar with these skills already. It’s important to be able to work out area and perimeter accurately to ensure you buy enough of each material (but not too much to avoid wastage).

Once your room is beautifully decorated, you’re going to need to plan the best layout for your furniture (and make sure it fits!). You may well use a scale drawing to achieve this.

By the end of this session you will be able to:

understand the difference between perimeter, area and volume and be able to calculate these for both simple and more complex shapes
know that volume is a measure of space inside a 3D object and calculate volumes of shapes in order to solve practical problems
draw and use a scale drawing or plan.

Download this video clip.Video player: bltl_a71_shape.mp4

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Transcript

[MUSIC PLAYING]

The task of designing things – like, gardens, for example – can be made much simpler if you have a basic understanding of shape and space. If you're putting up a fence around a garden and you need to calculate the total length of fencing you need, you'll save some money by ordering the correct amount. Or if you're putting fertiliser on a strange-shaped lawn, you can work out its surface area by breaking it down into smaller shapes.

If you wanted to dig a circular pond, you can use the radius of a circle to help calculate its area. And you can use the depth of a shape to calculate its volume. So when working out how far to dig, you don't have to depend on guesswork.

Understanding shapes and spaces can be surprisingly useful for something like gardening. But it won't help everything.

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1 Perimeter

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

As you will see from the picture below, perimeter is simply the distance around the outside of a shape or space. The shape you need to find the perimeter of could be anything from working out how much fencing you need to go around the outside of your garden to how much ribbon you need to go around the outside of a cake. Shapes or spaces with straight edges (rather than curved) are easiest to calculate so let’s begin by looking at these.

Figure _unit4.1.1 Figure 2 A perimeter fence

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Illustration of a perimeter fence around the outside edge (perimeter) of an area of garden.

Figure 2 A perimeter fence

1.1 Perimeter of simple shapes

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Figure _unit4.1.2 Figure 3 Finding the perimeter of simple shapes

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A rectangle, a triangle and a trapezium. The rectangle has sides labelled ‘10 cm’ and ‘6 cm’. The triangle has sides labelled ‘17 cm’, ‘12 cm’ and ‘12 cm’. The trapezium has sides labelled ‘12 cm’, ‘10 cm’, ‘18 cm’ and ‘10 cm’.

Figure 3 Finding the perimeter of simple shapes

In order to work out the perimeter of the shapes above, all you need to do is simply add up the total length of each of the sides.

Rectangle: 10 + 10 + 6 + 6 = 32 cm.
Triangle: 12 + 12 + 17 = 41 cm.
Trapezium: 10 + 12 + 10 + 18 = 50 cm.

When you give your answer, make sure you write the units in cm, m, km etc. One other important thing to note is that before you work out the perimeter of any shape, you must make sure all the measurements are given in the same units. If, for example, two lengths are given in cm and one is given in mm, you must convert them all to the same unit before you work out the total. It doesn’t usually matter which measurement you choose to convert but it’s wise to check the question first because sometimes you might be asked to give your answer in a specific unit.

Activity _unit4.1.1 Activity 1: Finding the perimeter

Work out the perimeters of each of the shapes below.

Remember to give units in your answer and to check that all measurements are in the same units before you begin to add. Carry out your calculations without using a calculator. Remember to check your answers against ours.

Figure _unit4.1.3 Figure 4 Calculating the perimeter – Question 1
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A rectangle. The horizontal side is 24 m. The vertical side is 18 m.
Figure 4 Calculating the perimeter – Question 1
Figure _unit4.1.4 Figure 5 Calculating the perimeter – Question 2
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A triangle. Two sides are 1.2 m. One side is 90 cm.
Figure 5 Calculating the perimeter – Question 2
Figure _unit4.1.5 Figure 6 Calculating the perimeter – Question 3
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A shape with 6 sides labelled ‘50 mm’, ‘10 cm’, ‘6 cm’, ‘62 mm’, ‘20 cm’ and ‘24 cm’.
Figure 6 Calculating the perimeter – Question 3

Answer

24 + 18 + 24 + 18 = 84 m.
If you worked in cm:
- 1.2 m = 120 cm.
  Therefore, the perimeter is 120 + 120 + 90 = 330 cm.
If you worked in m:
- 90 cm = 0.9 m.
  Therefore, the perimeter is 0.9 + 1.2 + 1.2 = 3.3 m.
50 mm = 5 cm, 62 mm = 6.2 cm.
Therefore, the perimeter is 24 + 5 + 10 + 6 + 6.2 + 20 = 71.2 cm.

Hopefully, you found working out the perimeters of these shapes fairly straightforward.

The next step on from shapes like the ones you have just worked with, is finding the perimeter of shapes where not all the lengths are given to you. In shapes such as a rectangle or regular shapes like squares (where all sides are the same length) this is a simple process. However, in a shape where the sides are not the same as each other, you have a little more work to do.

1.2 Perimeters of shapes with missing lengths

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Figure _unit4.1.6 Figure 7 Finding the perimeter when measurements are missing

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A shape with 6 sides. The 3 horizontal sides are 12 m, 20 m and 32 m. The vertical sides are 30 m and 17 m, and the remaining vertical side is labelled with a question mark.

Figure 7 Finding the perimeter when measurements are missing

Look at the shape above. You can see that one of the lengths is missing from the shape. How do you find the perimeter when you don’t have all the measurements? You cannot just assume that the missing length (yellow) is half of the red length, so how do you work it out? You’ll need to use the information that is given in the rest of the shape.

If you look at all the vertical lengths (red, yellow and green) you can see that we know the length of two of the three. You can also see that green + yellow = red, since the two shorter lengths put together would equal the same as the longest vertical length.

If 17 + ? = 30, then in order to find the missing length you must do 30 − 17 = 13.

The missing length is therefore 13 m. Now that you know this, you can work out the perimeter of the shape in the normal way.

30 + 12 + 13 + 20 + 17 + 32 = 124
So the perimeter of this shape is 124 m.

Let’s look at one more example before you try some on your own.

Figure _unit4.1.7 Figure 8 Finding the perimeter when a length is missing

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A shape with 6 sides. The 3 vertical sides are 15 m, 9 m and 6 m. The horizontal sides are 28 m, and 9 m, and the remaining horizontal side is labelled with a question mark.

Figure 8 Finding the perimeter when a length is missing

In the example above, you will see that there is again a missing length. This time the missing length (yellow) is a horizontal one and so you need to look at the other two horizontal lengths (red and green) in order to work out the missing side.

You can see that this time that red + green = yellow, since the missing length is the sum of the two shorter lengths. You can now do 9 + 28 = 37, so the missing length is 37 m.

Now that you know all the side lengths, you can work out the perimeter by finding the total of all the lengths.

15 + 9 + 6 + 28 + 9 + 37 = 104
So the perimeter of this shape is 104 m.

Activity _unit4.1.2 Activity 2: Perimeters and missing lengths

Work out the perimeter of the shape below.
Figure _unit4.1.8 Figure 9 Perimeters and missing lengths – Question 1
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A shape with 6 sides. The 3 horizontal sides are 12 m, 16 m and 28 m. The vertical sides are 27 m and 15 m, and the remaining vertical side is labelled with a question mark.
Figure 9 Perimeters and missing lengths – Question 1
You are redecorating your living room and need to replace the skirting board. The layout of the room is shown below. Skirting board can only be bought in lengths of 2 m.
How many lengths should you buy?
Figure _unit4.1.9 Figure 10 Perimeters and missing lengths – Question 2
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A shape with 6 sides. The 3 vertical sides are 5.2 m, 2.5 m and 2.7 m. The horizontal sides are 4.5 m, and 2.2 m, and the remaining horizontal side is labelled with a question mark.
Figure 10 Perimeters and missing lengths – Question 2

Answer

Firstly, you need to work out the missing length:
- 15 + 27 = 42 m
So the perimeter is:
- 42 + 12 + 27 + 16 + 15 + 28 = 140 m
Again, work out the missing length first:
- 4.5 − 2.2 = 2.3 m
Next, work out the perimeter of the room:
- 2.5 + 2.2 + 2.7 + 2.3 + 5.2 + 4.5 = 19.4 m
To see how many 2 m length of skirting you will need do:
- 19.4 ÷ 2 = 9.7
Since we can only buy whole lengths we need to round this up to 10 lengths of skirting.

Good work! You can now work out perimeters of simple and complex shapes, including shapes where there are missing lengths. There is just one more shape you need to consider – circles.

Whether it’s ribbon around a cake or fencing around a pond, it’s useful to be able to work out how much of a material you will need to go around the edge of a circular shape. The final part of your work on perimeter focusses on finding the distance around the outside of a circle.

1.3 Circumference of a circle

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

You may have noticed that a new term has slipped in to the title of this section. The term circumference refers to the distance around the outside of a circle – its perimeter. The perimeter and the circumference of a circle mean exactly the same, it’s just that when referring to circles you would normally use the term circumference rather than perimeter.

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Transcript

The circumference of a circle is the distance around its edge. That is, its perimeter. Before learning how to work out the circumference of a circle, look at these two key terms: diameter and radius. Radius, or r, is the distance from the centre of the circle to the edge. Diameter, or d, is the distance from one edge of the circle to the other, passing through the centre of the circle.

You can see the diameter is always double the radius. If you know either the radius or the diameter, you can always work out the other. And you can also work out the circumference. Here is the basic formula you need to work out the circumference of a circle: Circumference = pi x diameter. This can also be written as C = pi x d, or pi d.

Pi, or this symbol (which is the Greek letter used to represent it), is a constant number that is around the value of 3.142. It's a number that goes on for ever, so you tend to shorten it to the more manageable number of 3.14 or 3.142. In technical terms, pi is the number you get if you divide a circle's circumference by its diameter, and it's the same for every circle.

Let's look at an example of calculating circumference. This circle has a diameter of 5 centimetres. You can put this into the formula, so you get C = pi x 5. Look for the pi key on your calculator, or you can use the shortened version, 3.142. So the circumference, C = 3.142 x 5, which equals 15.71 centimetres.

Here's another example, but this time the radius is labelled. How would you calculate the circumference? Remember that the formula for circumference uses diameter, so you'll need to work this out first. Since the radius is 12 centimetres, the diameter will be double this. 12 x 2 = 24 centimetres, so d equals 24 centimetres. Now using the formula, circumference = 3.142 x 24, which equals 75.408.

Now, try the examples in the next activity.

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Activity _unit4.1.3 Activity 3: Finding the circumference

You have made a cake and want to decorate it with a ribbon.
The diameter of the cake is 15 cm. You have a length of ribbon that is 0.5 m long. Will you have enough ribbon to go around the outside of the cake?
Figure _unit4.1.10 Figure 11 A round chocolate cake
You have recently put a pond in your garden and are thinking about putting a fence around it for safety. The radius of the pond is 7.4 m.
What length of fencing would you require to fit around the full length of the pond? Round your answer up to the next full metre.
Figure _unit4.1.11 Figure 12 A round garden pond

Answer

d = 15 cm
Using the formula C = πd
- C = 3.142 × 15
- C = 47.13 cm
Since you need 47.13 cm and have ribbon that is 0.5 m (50 cm) long, yes, you have enough ribbon to go around the cake.
Radius = 7.4 m so the diameter is 7.4 × 2 = 14.8 m
Using the formula C = πd
C = 3.142 × 14.8
C = 46.5016 m which is 47 m to the next full metre.

You should now be feeling confident with finding the perimeter of all types of shapes, including circles. By completing Activity 3, you have also re-capped on using formulas and rounding.

The next part of this section looks at finding the area (space inside) a shape or space. As mentioned previously, this is incredibly useful in everyday situations such as working out how much carpet or turf to buy, how many rolls of wallpaper you need or how many tins of paint you need to give the wall two coats.

Summary

In this section you have learned:

that perimeter is the distance around the outside of a space or shape
how to find the perimeter of simple and more complex shapes
how to use the formula for finding the circumference of a circle.

2 Area

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

The area of a shape is the amount of space inside it. This applies to only two dimensional (flat) shapes. If we are dealing with a three dimensional shape, the space inside this is called the volume. Since perimeter is the measure of a length or distance, the units it is measured in are cm, m, km etc.

As area is a measure of space rather than length or distance, it is measured in square units. This could be square metres, square centimetres, square feet and so on. You will often see these written as cm², m², ft², etc. Over the next few pages you will learn how to find the area of simple shapes, compound shapes and circles.

2.1 Area of simple shapes

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

The simplest shapes to begin with when looking at area are squares and rectangles. If you look at the rectangle in Figure 13 you can see that it’s 6 cm long and 3 cm wide. If you count the squares, there are 18 of them. The area of the shape is 18 cm².

It is not always possible (or practical) to count the squares in a shape or space but it’s a useful illustration to help you understand what area is.

Extract _unit4.2.1

More practically, to find the area (A) of a square or a rectangle, you would multiply the length (l) by the width (w), so the formula would be:

Area = length × width or:
- A = lw (remember the multiplication sign is not normally written in a formula)

In the example below A = 6 × 3 = 18 cm².

Figure _unit4.2.1 Figure 13 Finding the area of a rectangle

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A rectangle on a square grid. The horizontal side is 6 cm and is six squares long. The vertical side is 3 cm and is 3 squares long.

Figure 13 Finding the area of a rectangle

Triangles are another shape that you can find the area of relatively simply. If you think of a triangle, it is really just half of a rectangle. This is easiest to see with a right-angled triangle as shown below. You can see that the actual triangle (in yellow) is simply a rectangle that has been diagonally cut in half.

In order to find the area of the triangle then, you simply multiply the base by the height (as you would for a rectangle) and then halve the answer.

Extract _unit4.2.2

Sometimes this is shown as the formula:

A = (b × h) ÷ 2
where b is the base of the triangle and h is the vertical height.

This formula may be written as:

For the triangle below then, you would do:

A = (5 × 4) ÷ 2
A = 20 ÷ 2
A = 10 cm²

Figure _unit4.2.2 Figure 14 Finding the area of a right-angled triangle

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A triangle on a square grid. The vertical side is 4 cm and is 4 squares long. The horizontal side is 5 cm and is 5 squares long.

Figure 14 Finding the area of a right-angled triangle

This formula remains the same for any triangle. Take a look at the triangle below. It’s a bit less obvious than with the example above but if you were to take the two yellow sections away and put them together, you would end up with a shape exactly the same size as the orange triangle.

Extract _unit4.2.3

The area of this triangle can be found in the same way as the previous one:

A = (b × h) ÷ 2
A = (4 × 7) ÷ 2
A = 28 ÷ 2
A = 14 cm²

Figure _unit4.2.3 Figure 15 Finding the area of a triangle

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A triangle positioned on a square grid. A line from the triangle’s base to its tip is labelled ‘7 cm’ and is 7 squares long. The base of the triangle is 4 cm and is 4 squares long.

Figure 15 Finding the area of a triangle

Another shape you may need to find the area of is the trapezium. You will need to use a simple formula for this shape (don’t panic when you see it, it looks scary but it really is quite easy to use!)

A trapezium looks like any of the shapes below.

Figure _unit4.2.4 Figure 16 Examples of trapezium shapes

In order to work out the area of a trapezium you just need to know the vertical height and the length of the top and bottom sides. Traditionally, the length of the top is called ‘a’, the length on the bottom is ‘b’ and the vertical height is ‘h’. Once this is clear you can then use the formula:

Extract _unit4.2.4

Figure _unit4.2.5 Figure 17 Dimensions of a trapezium

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A trapezium. The length of the top side is labelled ‘a’. The length of the bottom side is labelled ‘b’. The length between the top and bottom sides (height) is labelled ‘h’.

Figure 17 Dimensions of a trapezium

Let’s take a look at an example on how to work out the area of the trapezium below. We can see that the top length (a) = 12 cm. The bottom length (b) = 20 cm, and the height (h) = 13 cm.

Extract _unit4.2.5

Using these values and the formula:

A =
A =
A =
A = 208 cm²

Figure _unit4.2.6 Figure 18 Finding the area of a trapezium

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A trapezium. The length of the top side is 12 cm. The length of the bottom side is 20 cm. The length between the top and bottom sides (height) is 13 cm.

Figure 18 Finding the area of a trapezium

Now that you have seen how to work out the area of several basic shapes, it’s time to put your skills to the test. Have a go at the activity below. Don’t forget that, just as with perimeter, before you can begin any calculations you must make sure all measurements are in the same units.

Activity _unit4.2.1 Activity 4: Finding the area

Work out the area of each of the shapes below.

Figure _unit4.2.7 Figure 19 Finding the area – Question 1
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A rectangle. The horizontal side is 1.6 m. The vertical side is 95 cm.
Figure 19 Finding the area – Question 1

Answer

You need to convert the measurements into the same units before you can work out the area.
If working in cm:
- 1.6 m = 160 cm, so
  A = 160 × 95 = 15 200 cm²
If working in m:
- 95 cm = 0.95 m, so
  A = 1.6 × 0.95 = 1.52 m²

Figure _unit4.2.8 Figure 20 Finding the area – Question 2
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A right-angled triangle. The vertical side is 17 cm. The horizontal side is 30 cm. The third side is 46 cm.
Figure 20 Finding the area – Question 2

Answer

The two measurements we need for the triangle are the base (30 cm) and the vertical height (17 cm). Don’t be fooled by the diagonal length of 46 cm, you do not need it for the area!
- A = (b × h) ÷ 2
- A = (30 × 17) ÷ 2
- A = 510 ÷ 2
- A = 255 cm²

Figure _unit4.2.9 Figure 21 Finding the area – Question 3
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A triangle. The length of the base is 60 mm. The length from the base to the tip (height) is 14 cm.
Figure 21 Finding the area – Question 3

Answer

You need to convert the measurements into the same units before you can work out the area.
If working in mm: 14 cm = 140 mm
- A = (b × h) ÷ 2
- A = (60 × 140) ÷ 2
- A = 8400 ÷ 2
- A = 4200 mm²

If working in cm: 60 mm = 6 cm
- A = (b × h) ÷ 2
- A = (6 × 14) ÷ 2
- A = 84 ÷ 2
- A = 42 cm²

Figure _unit4.2.10 Figure 22 Finding the area – Question 4
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A trapezium. The length of the top side is 8 cm. The length of the bottom side is 15 cm. The length between the top and bottom sides (height) is 9 cm.
Figure 22 Finding the area – Question 4

Answer

Top length (a) = 8 cm

Bottom length (b) = 15 cm
Height (h) = 9 cm
Using the formula for the area of a trapezium:
- A =
- A =
- A =
- A = 103.5 cm²

Now that you’ve mastered finding the area of basic shapes, it’s time to look at compound shapes.

A compound shape is just a shape that is made from more than one basic shape. Rarely will you find a floor space, garden area or wall that is perfectly rectangular. More often than not it will be a combination of shapes. The good news is that, to find the area of compound shapes, you simply split them up into their basic shapes, find the area of each of these, and then add them up at the end!

2.2 Area of compound shapes

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Take a look at the shape below; this is an example of a compound shape. Whilst you cannot find the area of this shape as it is by using a formula as you have done previously, you can split it into two basic shapes (rectangles) and then use your existing knowledge to work out the area of each of these shapes.

Figure _unit4.2.11 Figure 23 Finding the area of a compound shape

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A compound ‘L’ shape with 6 sides. The length along the base of the shape is 15 cm. The other horizontal sides are 5 cm and 10 cm. The vertical sides are 9 cm, 5 cm and 4 cm.

Figure 23 Finding the area of a compound shape

You should be able to see that you can split this shape into two rectangles. It does not matter which way you split it – you will get the same answer at the end.

You could split it like this:

Figure _unit4.2.12 Figure 24 Splitting a compound shape horizontally to find the area

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The compound shape from Figure 1 split horizontally into two rectangles. The sides of the left-hand rectangle are 5 cm and 9 cm. The sides of the right-hand rectangle are 10 cm and 4 cm.

Figure 24 Splitting a compound shape horizontally to find the area

Extract _unit4.2.6

You now have two rectangles. To work out the area of rectangle ①, you do A = 9 × 5 = 45 cm².

To work out the area of rectangle ②, you do A = 10 × 4 = 40 cm².

Now that you have the area of both rectangles, simply add them together to find the area of the whole shape:

45 + 40 = 85 cm²

You need to be careful that you are using the correct measurements for the length and width of each rectangle (the measurements in red). In this example, the lengths of 15 cm and 5 cm (in black) are not required.

Alternatively, you could split the shape like this:

Figure _unit4.2.13 Figure 25 Splitting a compound shape vertically to find the area

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The compound shape from Figure 1 split vertically into two rectangles. The sides of the top rectangle are 5 cm and 5 cm. The sides of the bottom rectangle are 15 cm and 4 cm.

Figure 25 Splitting a compound shape vertically to find the area

Extract _unit4.2.7

Again, you now have two rectangles. To work out the area of rectangle ①, you do A = 5 × 5 = 25 cm².

To work out the area of rectangle ②, you do A = 15 × 4 = 60 cm².

Now that you have the area of both rectangles, simply add them together to find the area of the whole shape:

25 + 60 = 85 cm²

Again, you need to be careful that you are using the correct measurements for the length and width of each rectangle (the measurements in red). In this example, the lengths of 9 cm and 10 cm (in black) are not required.

You will notice that regardless of which way you choose to split the shape, you arrive at the same answer of 85 cm².

The best way for you to practise this skill is to try a few examples for yourself. Have a go at the activity below and then check your answers.

Activity _unit4.2.2 Activity 5: Finding the area of compound shapes

Work out the area of the shapes below.

Figure _unit4.2.14 Figure 26 Finding the area of compound shapes – Question 1
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A compound ‘L’ shape with 6 sides. The height of the shape is 15 cm. The other vertical sides are 9 cm and 6 cm. The horizontal sides are 4 cm, 8 cm and 12 cm.
Figure 26 Finding the area of compound shapes – Question 1

Answer

The area of the whole shape is 108 cm²
Depending on how you split the shape you may have done:
- 6 × 8 = 48 cm²
- 15 × 4 = 60 cm²
- 48 + 60 = 108 cm²
or:

- 12 × 6 = 72 cm²
- 9 × 4 = 36 cm²
- 72 + 36 = 108 cm²

Figure _unit4.2.15 Figure 27 Finding the area of compound shapes – Question 2
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A compound ‘L’ shape with 6 sides. The length along the base of the shape is 20 cm. The other horizontal sides are 12 cm and a question mark. The vertical sides are 9 cm, 4 cm and a question mark.
Figure 27 Finding the area of compound shapes – Question 2

Box _unit4.2.1

Hint: You’ll need to find some missing lengths on this shape before you can work out the area.

Answer

The missing vertical length is 13 cm (9 cm + 4 cm) and horizontal length is 8 cm (20 cm − 12 cm). The area of the whole shape is 212 cm².
Depending on how you split the shape you may have done:
- 13 × 8 = 104 cm²
- 12 × 9 = 108 cm²
- 104 + 108 = 212 cm²
or:
- 20 × 9 = 180 cm²
- 4 × 8 = 32 cm²
- 180 + 32 = 212 cm²

Now that you can calculate the area of basic and compound shapes, there is just one other shape you will practise finding the area of: circles. Similarly to finding the perimeter of a circle, you’ll need to use a formula involving the Greek letter π.

2.3 Area of a circle

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Figure _unit4.2.16 Figure 28 The circumference and area of a circle

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A circle annotated with the formulas ‘C = πd’ and ‘A = πr²’.

Figure 28 The circumference and area of a circle

You have already practised using the formula to find the circumference of a circle. You will now look at using the formula to find the area of one.

Extract _unit4.2.8

To find the area of a circle you need to use the formula:

Area of a circle = pi × radius²
This can also be written as:
- A = πr²
- where:
- A = area
- π = pi
- r = radius
- r² means r squared.

Box _unit4.2.2

Hint: Remember that when you square a number you simply multiply it by itself, radius² is therefore simply radius × radius.

Let’s look at an example.

Figure _unit4.2.17 Figure 29 The radius of a circle

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A circle with a line from the centre point to the outside edge labelled ‘8 cm’.

Figure 29 The radius of a circle

Extract _unit4.2.9

In the circle above you can see that the radius is 8 cm. For these tasks we will use the figure of 3.142 for π.

To find the area of the circle we need to do:

A = πr²
A = 3.142 × 8 × 8
A = 201.088 cm²

Before you try some on your own, let’s take a look at one further example.

Figure _unit4.2.18 Figure 30 The diameter of a circle

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A circle with a line labelled ‘12 cm’. The line is from the outside edge to the opposite outside edge and passes through the centre point.

Figure 30 The diameter of a circle

Extract _unit4.2.10

This circle has a diameter of 12 cm. In order to find the area, you first need to find the radius. Remember that the radius is simply half of the diameter and so in this example radius = 12 ÷ 2 = 6 cm.

We can now use:

A = πr²
A = 3.142 × 6 × 6
A = 113.112 cm²

Try a couple of examples for yourself before moving on to the next part of this section.

Activity _unit4.2.3 Activity 6: Finding the area of a circle

Find the area of the circle shown below. Give your answer to one decimal place.

Figure _unit4.2.19 Figure 31 Finding the area of a circle – Question 1
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A circle with a line from the centre point to the outside edge labelled ‘4 m’.
Figure 31 Finding the area of a circle – Question 1

Find the area of the circle shown below. Give your answer to 1 decimal place.

Figure _unit4.2.20 Figure 32 Finding the area of a circle – Question 2
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A circle with a straight line going from left to right and through the centre point of the circle labelled ‘16 m’.
Figure 32 Finding the area of a circle – Question 2

You are designing a mural for a local school and need to decide how much paint you need. The main part of the mural is a circle with a diameter of 10 m as shown below. Each tin of paint will cover an area of 5 m². You will need to use two coats of paint. How many tins of paint should you buy?

Figure _unit4.2.21 Figure 33 Finding the area of a circle – Question 3
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A circle with a straight line going from left to right and through the centre point of the circle labelled ‘10 m’.
Figure 33 Finding the area of a circle – Question 3

Answer

To find the area of the circle you need to do:
- A = πr²
- A = 3.142 × 4 × 4
- A = 50.272 m²
- A = 50.3 m² to 1 d. p.
You need to find the radius of the circle first. Since the diameter of the circle is 16 cm, the radius is 16 cm ÷ 2 = 8 cm. Now you can find the area:
- A = πr²
- A = 3.142 × 8 × 8
- A = 201.088 cm²
- A = 201.1 cm² to one d.p.
You need to find the area of the circle first. Since the diameter of the circle is 10 m, the radius is 10 m ÷ 2 = 5 m.
Now you can find the area of the circle:
- A = πr²
- A = 3.142 × 5 × 5
- A = 78.55 m²
So the area of the circle you need to paint is 78.55 m²
Since you need to give 2 coats of paint, you need to double this number:
- 78.55 × 2 = 157.1 m²
You now need to work out how many tins of paint you need. As one tin of paint covers 5 m² you need to do:
- 157.1 ÷ 5 = 31.42 tins
Since you must buy whole tins of paint, you will need to buy 32 tins.

You have now learned all you need to know about finding the area of shapes! The last part of this section is on finding the volume of solid shapes – or three-dimensional (3D) shapes.

Summary

In this section you have learned:

that area is the space inside a two-dimensional (2D) shape or space
how to find the area of rectangles, triangles, trapeziums and compound shapes
how to use the formula to find the area of a circle.

3 Volume

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

The volume of a shape is how much space it takes up. You might need to calculate the volume of a space or shape if, for example, you wanted to know how much soil to buy to fill a planting box or how much concrete you need to complete your patio.

Figure _unit4.3.1 Figure 34 A volume cartoon

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A volume cartoon. The caption reads ‘What if someone hits the mute button?’

Figure 34 A volume cartoon

You’ll need your area skills in order to calculate the volume of a shape. In fact, as you already know how to calculate the area of most shapes, you are one simple step away from being able to find the volume of most shapes too!

Download this video clip.Video player: bltl_4_3_3d_shapes.mp4

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Transcript

In this video, you'll look at how to calculate the volume of prisms. A prism is a 3D shape where the cross-section, or slice, of the shape is the same all the way through. You can see that a slice of any of these shapes would remain the same wherever you cut it. To calculate the volume of any prism, all you need to do is find the area of the cross-section and then multiply this area by the length of the prism.

In this example, you firstly need to calculate the area of the rectangular cross-section. 12 times 10 gives an area of 120 centimetres squared. Now you know the area of the cross-section, all you need to do is multiply it by the length of the shape: 5 centimetres. Volume equals 120 times 5, which equals 600 centimetres cubed.

Let's try again with a different shape. Here, the cross-section is the triangle. You'll recall from your earlier work that to find the area of a triangle, you use this formula. So the area of this cross-section is 6 times 5, divided by 2. That is 30 divided by 2. This gives an area of 15 centimetres squared. Now, multiply the area of the cross-section by the length of the shape to find the volume. 15 times 12 gives a volume of 180 centimetres cubed. Look at a couple more examples of calculating the volume of prisms, before having a go yourself.

End transcript

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Case study _unit4.3.1 Example: Calculating volume

Figure _unit4.3.2 Figure 35 Calculating volume

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A 3D L shape split into two rectangles. The depth of the whole shape is 10 cm and the width of the whole shape is 9 cm. The other lengths of the left-hand rectangle are 7 cm (vertical) and 4 cm (horizontal). The other lengths of the right-hand rectangle are 2 cm (vertical) and 5 cm (horizontal).

Figure 35 Calculating volume

The cross-section on this shape is the L shape on the front. In order to work out the area you’ll need to split it up into two rectangles as you practised in the previous part of this section.

Extract _unit4.3.1

Rectangle 1 = 7 × 4 = 28 cm²

Rectangle 2 = 5 × 2 = 10 cm²

Area of cross-section = 28 + 10 = 38 cm²

Now you have the area of the cross-section, multiply this by the length to calculate the volume.

Extract _unit4.3.2

V = 38 × 10 = 380 cm³

Case study _unit4.3.2 Example: Calculating cylinder volume

The last example to look at is a cylinder. The cross-section of this shape is a circle. You’ll need to use the formula to find the area of a circle in the same way you did in the previous part of this section.

Figure _unit4.3.3 Figure 36 Calculating cylinder volume

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A cylinder. The radius of the base is 8 cm. The height is 15 cm.

Figure 36 Calculating cylinder volume

You can see that the circular cross-section has a radius of 8 cm. To find the area of this circle, use the formula:

Extract _unit4.3.3

A = πr²

A = 3.142 × 8 × 8

A = 201.088 cm²

Now you have the area, multiply this by the length of the cylinder to calculate the volume.

Extract _unit4.3.4

V = 201.088 × 15 = 3016.32 cm³

Now try the following questions. Please carry out the calculations without a calculator. You can double check with a calculator if needed and remember to check your answers against ours.

Activity _unit4.3.1 Activity 7: Volume

Find the volume of the following shapes without using a calculator. The shapes are not drawn to scale.

Figure _unit4.3.4 Figure 37 Rectangular prism (cuboid)
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A rectangular prism (cuboid). The length is 12 cm, the height is 5 cm and the depth is 3 cm.
Figure 37 Rectangular prism (cuboid)

Answer

The area of the rectangular cross-section is:
- 5 × 12 = 60 cm²
To get the volume, you now need to multiply the area of the cross-section by the length of the shape:
- 60 × 3 = 180 cm³

Figure _unit4.3.5 Figure 38 Circular prism (cylinder)
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A circular prism (cylinder). The diameter is 12 cm, the height is 35 cm.
Figure 38 Circular prism (cylinder)

Answer

You can see that the circular cross-section has a diameter of 20 cm. To find the area of this circle, you need to find the radius first:
- radius = 20 ÷ 2 = 10 cm
To find the area of the circular cross-section, use the formula:
- A = πr²
- A = 3.142 × 10 × 10
- A = 314.2 cm²
Now you have the area, multiply this by the length of the cylinder to calculate the volume:
- V = 314.2 × 35 = 10 997 cm³

Figure _unit4.3.6 Figure 39 Triangular prism
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A triangular prism. The base is 1.5 m, the height is 1.5 m and the depth is 3 m.
Figure 39 Triangular prism

Answer

The cross-section on this shape is the triangle at the front of the shape. To find the area of a triangle you use the formula:
- A = (b × h) ÷ 2
In this case then:
- A = (1.5 × 1.5) ÷ 2
- A= 2.25 ÷ 2
- A = 1.125 m²
Now you know the area of the cross-section, you multiply it by the length of the shape to find the volume (V):
- V = 1.125 × 3 = 3.375 m³

Figure _unit4.3.7 Figure 40 Trapezoidal prism
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A 3D trapezium. The length of the top side is 6 cm. The length of the bottom side is 8 cm. The length between the top and bottom sides (height) is 5 cm. The depth is 20 cm.
Figure 40 Trapezoidal prism

Box _unit4.3.1

Hint: Go back to the section on area and remind yourself of the formula to find the area of a trapezium.

Answer

The cross-section is a trapezium so you will need the formula:
- A =
- A =
- A =
- A =
- A = 35 cm²
Now you have the area of the cross-section, multiply it by the length of the prism to calculate the volume:
- V = 35 × 20 = 700 cm³

Summary

In this section you have learned:

that volume is the space inside a 3D shape or space
how to find the volume of prisms such as cuboids, cylinders and triangular prisms.

4 How many will fit?

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

You may need to work out how many tiles to buy to tile an area of wall or how many tins you can pack in a box. We will use the example below to illustrate.

Case study _unit4.4.1 Example 1: Tiling a wall

You are going to buy tiles measuring 0.75 m by 0.75 m to tile this area of wall:

Figure _unit4.4.1 Figure 41 Wall measurements for tiling

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A brick wall with a width measurement of 5.25 m and a height measurement of 3 m.

Figure 41 Wall measurements for tiling

How many tiles do you need to buy?

The easiest way to tackle a question such as this is to work out how many tiles will fit across the wall and how many will fit down the wall. You can then work out how many you need altogether.

The length across the wall is 5.25 m so we can divide by the length of the tile to work out how many will fit:

5.25 ÷ 0.75 = 7 tiles across

You may have preferred to convert the measurements into centimetres to avoid dividing by a decimal:

1 m = 100 cm, so
5.25 × 100 = 525 cm
0.75 × 100 = 75 cm
525 ÷ 75 = 7 tiles (you can see that the number of tiles is the same.)

Down the wall the measurement is 3 m, so down the wall you will fit:

3 ÷ 0.75 = 4 tiles (you may have converted to cm again and done 300 ÷ 75 = 4 tiles)

If 7 tiles will fit across the wall and 4 will fit down then altogether you will need:

7 × 4 = 28 tiles.

Case study _unit4.4.2 Example 2: Packing calculations

A shop is packing tins of varnish into boxes for delivery. The tins are cylindrical with a diameter of 7.5 cm and a height of 10 cm. The packing box is 60 cm long, 45 cm wide and 30 cm high.

How many tins can be packed into each box?

Figure _unit4.4.2 Figure 42 Dimensions of varnish tins and packing box

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A tin of varnish is on the left of the image. The diameter of the tin is labelled 7.5 cm and the height is labelled 10 cm. The right of the image has a different coloured background and shows a storage box. The length of the storage box is labelled 60 cm, the height 30 cm and the depth 45 cm.

Figure 42 Dimensions of varnish tins and packing box

The easiest way to tackle this type of question is to work out how many tins you can fit in a single layer first and then to work out how many layers you can have. To work out how many tins will fit in a single layer use the same method you used above for working out how many tiles would fit.

The box is 60 cm long and tins are 7.5 cm wide:

60 ÷ 7.5 = 8 so 8 tins will fit across the box.

Hint: make sure you are working with the diameter and not the radius of the cylinder.

The box is 45 cm wide:

45 ÷ 7.5 = 6
so 6 tins would fit.

This would give 6 rows of 8 tins:

8 × 6 = 48
so 48 tins would fit in 1 layer.

Next you need to work out how many layers you could fit. The tins are 10 cm high and the box is 30 cm high:

30 ÷ 10 = 3
so the tins can be stacked 3 high:
- 3 × 48 = 144
The box would hold 144 tins.

Now try the following questions. Please carry out the calculations without a calculator. You can double check with a calculator if needed and remember to check your answers against ours.

Activity _unit4.4.1 Activity 8: How many will fit?

You want to re-carpet your floor using carpet tiles. Your floor measures 4.5 m by 6 m. The tiles you like measure 0.5 m by 0.5 m and cost £1.89 per tile.
- a.How many tiles will you need to buy?
- b.How much will they cost altogether?

Answer

- a.You need to work out how many tiles will fit across the length of the floor and across the width of the floor.
  Length
  The floor is 6 m long. Each tile is 0.5 m by 0.5 m so to work out how many will fit do:
       6 ÷ 0.5 = 12 tiles
  You could also convert to centimetres:
       6 m = 6 × 100 = 600 cm
       0.5 m = 0.5 × 100 = 50 cm
       600 ÷ 50 = 12 tiles
  Width
  The floor is 4.5 m wide. Each tile is 0.5 m by 0.5 m so to work out how many will fit do:
       4.5 ÷ 0.5 = 9 tiles
  You could also convert to centimetres:
       4.5 m = 4.5 × 100 = 450 cm
       0.5 m = 0.5 × 100 = 50 cm
       450 ÷ 50 = 9 tiles
  So altogether you need:
       12 × 9 = 108 tiles
- b.Each tile costs £1.89 and you need 108 tiles so the total cost of the tiles will be:
  1.89 × 108 = £204.12

How many 25 cm square tiles would you need to tile the floor area below?
Figure _unit4.4.3 Figure 43 Area of floor for tiling
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An irregular area of floor that has measurements next to most of the lengths. The measurements are: bottom length = 4.5 m; left length = 3 m; top length = 2 m; right length = 1.5 metres. All the angles are right angles and two sides don’t have measurements.
Figure 43 Area of floor for tiling

Answer

You need to split the floor up into 2 rectangles. You could do this in a couple of different ways.
First method for splitting the floor as follows:
Figure _unit4.4.4 Figure 44 First method for splitting the floor
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The irregular shape in Figure 43 with the same measurements is split into two rectangles. The line that splits the rectangles is drawn vertically.The rectangle on the left is labelled ‘1’ and the one on the right is labelled ‘2’.
Figure 44 First method for splitting the floor
Rectangle 1
Rectangle ① is 3 m by 2 m. Convert the dimensions into centimetres:
- 3 m × 100 = 300 cm
- 2 m × 100 = 200 cm
The tiles are 25 cm square (25 cm by 25 cm) so you will need:
- 300 ÷ 25 = 12
- 200 ÷ 25 = 8
- so for this part of the floor you will need:
- 12 × 8 = 96 tiles.
Rectangle 2
Rectangle ② is 1.5 m by 2.5 m (4.5 m − 2 m = 2.5 m). Convert the dimensions into centimetres:
- 1.5 m × 100 = 150 cm
- 2.5 m × 100 = 250 cm
The tiles are 25 cm square (25 cm by 25 cm) so you will need:
- 150 ÷ 25 = 6
- 250 ÷ 25 = 10
- so for this part of the floor you will need:
- 6 × 10 = 60 tiles.
So altogether you will need:
- 60 + 96 = 156 tiles to tile the floor.
You may have split the floor area up differently.
Second method for splitting the floor as follows:
Figure _unit4.4.5 Figure 45 Second method for splitting the floor
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The irregular shape in Figure 43 with the same measurements is split into two rectangles. The line that splits the rectangles is drawn horizontally. The rectangle on the top is labelled ‘1’ and the one on the bottom is labelled ‘2’.
Figure 45 Second method for splitting the floor
Rectangle 1
Here rectangle ① is 1.5 m by 2 m (3 − 1.5 m = 2 m). Convert the dimensions into centimetres:
- 1.5 m × 100 = 150 cm
- 2 m × 100 = 200 cm
The tiles are 25 cm square (25 cm by 25 cm) so you will need:
- 150 ÷ 25 = 6
- 200 ÷ 25 = 8
- so for this part of the floor you will need:
- 6 × 8 = 48 tiles
Rectangle 2
Here rectangle ② is 1.5 m by 4.5 m. Convert the dimensions into centimetres:
- 1.5 m × 100 = 150 cm
- 4.5 m × 100 = 450 cm
The tiles are 25 cm square (25 cm by 25 cm) so you will need:
- 150 ÷ 25 = 6
- 450 ÷ 25 = 18
- so for this part of the floor you will need:
- 6 × 18 = 108 tiles
So altogether you will need:
- 48 + 108 = 156 tiles to tile the floor.

You are packing boxes of chocolates. Each chocolate box measures 23 cm long by 15 cm wide by 4 cm high and you are packing them into a storage box measuring 48 cm long by 30 cm wide by 34 cm high.
What is the maximum number of chocolate boxes that can be packed in one storage box?
Hint: The chocolate boxes need to be packed flat (horizontally) but you may want to try turning the chocolate box around.
Figure _unit4.4.6 Figure 46 Chocolate boxes to fit into a storage box
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A chocolate box is on the left of the image with dimensions: length = 23 cm; height = 4 cm; depth = 15 cm. The storage box is on the right with dimensions: length = 48 cm; height = 34 cm; depth = 30 cm.
Figure 46 Chocolate boxes to fit into a storage box

Answer

The chocolate boxes need to be packed flat (horizontally). However, the boxes could be packed in 2 different ways:
Figure _unit4.4.7 Figure 47 Method 1 and 2 for packing the chocolate boxes
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The chocolate box from Figure 46 is shown twice. The label ‘Method 1’ is at the top left of the image and under the chocolate box is shown with the longest side facing the viewer. The label ‘Method 2’ is at the top right of the image with the chocolate box turned 90 degrees underneath it.
Figure 47 Method 1 and 2 for packing the chocolate boxes
Remember the dimensions of the storage box are as follows:
Figure _unit4.4.8 Figure 48 Storage box dimensions
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The storage box is with dimensions is repeated: length = 48 cm; height = 34 cm; depth = 30 cm.
Figure 48 Storage box dimensions
Method 1
If you packed the chocolate boxes this way the storage box is 48 cm long and the chocolate boxes are 23 cm long so the calculation would be:
- 48 ÷ 23 = 2.1 (rounded to one d.p.)
- so only 2 chocolate boxes will fit.
The storage box is 30 cm wide and the chocolate boxes are 15 cm wide so:
- 30 ÷ 15 = 2
- so 2 chocolate boxes would fit.
This would give 2 rows of 2 chocolate boxes:
- 2 × 2 = 4
- so 4 chocolate boxes would fit in 1 layer.
Next you need to work out how many layers you could fit. The storage box is 34 cm high and the chocolate boxes are 4 cm high:
- 34 ÷ 4 = 8.5
- so 8 chocolate boxes will fit.
The chocolate boxes can be stacked 8 high so:
- 8 × 4 = 32
If using Method 1 the storage box would hold 32 chocolate boxes.
Method 2
If you packed the chocolate boxes this way the calculation would be as follows:
- 48 ÷ 15 = 3.2 so 3 boxes will fit
- 30 ÷ 23 = 1.3 (rounded to one d.p.) so only 1 will fit
In one layer, you could fit 3 × 1 = 3 chocolate boxes.
Next you need to work out how many layers you could fit. The storage box is 34 cm high and the chocolate boxes are 4 cm high:
- 34 ÷ 4 = 8.5
- so only 8 chocolate boxes will fit.
The chocolate boxes can be stacked 8 high:
- 8 × 3 = 24
If using Method 2 the storage box would hold 24 chocolate boxes.
Therefore, the maximum number of chocolate boxes that could fit into the storage box is 32 and you would have to pack them the first way (Method 1) to achieve this.

A factory producing jars of jam packs the jars into boxes that measure 42 cm long, 42 cm wide and 45 cm high. The jars of jam have a radius of 3.5 cm and a height of 11 cm.
How many jars of jam can be packed into a box? The jars must be packed upright.
Figure _unit4.4.9 Figure 49 Jam jars (left) to fit into a packaging box (right)
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A jar labelled ‘Blueberry Jam’ is on the left with dimensions labelled: radius = 3.5 cm; height = 11 cm. The packaging box is on the right with dimensions: length = 42 cm; height = 45 cm; depth = 42 cm.
Figure 49 Jam jars (left) to fit into a packaging box (right)

Answer

First you need to calculate the diameter (width) of the jars. The radius of the jars is 3.5 cm:
- 3.5 × 2 = 7
- so the diameter of the jars is 7 cm.
The box is 42 cm wide and jars are 7 cm wide:
- 42 ÷ 7 = 6
- so 6 jars will fit across the box.
The length of the box is also 42 cm so you know that there will be 6 rows of 6 jars on the bottom layer.
- 6 × 6 = 36
- so 36 jars will fit in 1 layer.
Next you need to work out how many layers you could fit. The jars are 11 cm high, and the box is 45 cm high:
- 45 ÷ 11 = 4.09 to two d.p.
- so the tins can be stacked 4 high.
- 4 × 36 = 144
- So 144 jars of jam would fit in each box.

The final part of his section will look at scale drawings and plans. It will require your previous knowledge of ratio as scale drawings are really just another application of ratio. The good news then, is that you already know how to do it!

Summary

In this section you have:

learned to calculate how many smaller shapes or items will fit into larger areas or spaces.

5 Scale drawings and plans

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

If you completed Everyday maths 1, you will be familiar with the idea of scale. Scales are found on drawings, plans and maps and they are often written with the units indicated. Let’s look at an example.

A football pitch is drawn to the scale of 1 cm to 5 m.

Figure _unit4.5.1 Figure 50 A football pitch drawn at 1 cm to 5 m

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A football pitch with a label at the top saying ‘Football pitch scale: 1 cm to 5 m’.

Figure 50 A football pitch drawn at 1 cm to 5 m

This means that every 1 cm measured on the plan is 5 m in real life.

If the plan is drawn with the length being 18 cm and the width being 9 cm, what are the dimensions of the football pitch in real life?

Extract _unit4.5.1

Write down the scale first:

1 cm to 5 m

You know the drawing dimensions so you need to work with these one at a time. Let’s start with the length of 18 cm:

If the scale is 1 cm to 5 m then
18 cm = ? m.
If you have been given the drawing measurement and need to know the real life measurement, you multiply:
18 × 5 = 90 m
so the length is 90 m.

Box _unit4.5.1

Note: If you have been given the actual measurement and need to find the drawing measurement you would need to divide.

Now you can work out the width measurement:

If the scale is 1 cm to 5 m then
9 cm = ? m.
Again, you need to multiply:
9 × 5 = 45 m
so the width is 45 m.

A lot of scales are written differently, without the units indicated. The scale of 1 cm to 5 m could also be written as 1:500.

This is the scale expressed as a ratio and it is independent of any units. A scale of 1:500 means that the actual real-life measurements are 500 times greater than those on the plan or map. This means that it does not matter whether you take the measurements on the plan in millimetres (mm), centimetres (cm) or metres (m) – the measurements will be 500 times as much in real life.

To write a scale as a ratio, you often have to convert. Let’s look at the football pitch example again:

Extract _unit4.5.2

1 cm to 5 m

At the moment, the units of the scale are different. The plan side is given in centimetres (cm) and the real-life side is given in metres (m).

To express this as a ratio, you need to convert both sides to the same units. It is usually easiest to convert the real-life side of the scale into the same unit as the drawing side, so in this case it is easiest to convert 5 m into cm:

5 × 100 = 500 cm

So you can now write the scale as a ratio:

1:500

It is standard to try to write the ratio in the simplest form possible, ideally with a single unit (a ‘1’) on the drawing side of the ratio. This will make any calculations you do using the scale easier.

Now have a go a converting scales to ratios.

Activity _unit4.5.1 Activity 9: Writing a scale as a ratio

Rewriting these scales as a ratio in their simplest form:

1 cm to 2 m
2 cm to 5 m
10 mm to 20 m
1 cm to 1 km
5 cm to 2 km

Answer

It is easiest to change the 2 m into cm:
- 2 × 100 = 200 cm so the scale expressed as a ratio would be 1:200.
It is easiest to change the 5 m into cm:
- 5 × 100 = 500 cm so the scale expressed as a ratio could be written as:
- 2:500
However, we usually try to get the drawing side of the ratio down to a single unit (1) to make calculations easier. Therefore, you need to simplify the ratio. To do this here, divide both sides by 2:
- 2 ÷ 2 = 1
- 500 ÷ 2 = 250 so the scale can be written as:
- 1:250
It is easiest to change the 20 m into mm. It might be easiest to do this in stages:
- Convert to cm first –
  1 m = 100 cm so 20 × 100 = 2000 cm
- Now convert to mm –
  1 cm = 10 mm so 2000 × 10 = 20 000 mm
- This makes the scale:
- 10:20 000
- This can be simplified by dividing both sides by 10 to get:
- 1:2000
Change the 1 km into cm. Again, this will be easiest to do in stages:
- Convert to m first –
  1 km = 1000 m so 1 × 1000 = 1000 m
- Now convert to cm –
  1 m = 100 cm so 1000 × 100 = 100 000 cm
- This means the scale should be written as:
- 1:100 000
Change the 2 km into cm. In stages this can be done as follows:
- Convert to m first –
  1 km = 1000 m so 2 × 1000 = 2000 m
- Now convert to cm –
  1 m = 100 cm so 2000 × 100 = 200 000 cm
- This makes the scale:
- 5:200 000
- This can be simplified by dividing both sides by 5 to get:
- 1:40 000

Now you will look at using ratio scales to work out measurements.

5.1 Scale drawing method and problems

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

First watch the scale drawings example video.

Download this video clip.Video player: bltl_4_4_trapezium.mp4

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Transcript

You have already developed skills in using measures and ratio. Scale drawings are just another application of this knowledge. So the good news is you already know how to do it. With this topic, it's best to dive straight in with an example.

Ahmed is going to make some home improvements. He wants to build a conservatory. Here is Ahmed's sketch of the conservatory. He uses a scale of 1 to 50. You need to draw an accurate scale drawing of the side view, using Ahmed's scale.

The scale, or ratio, we are working with is 1 to 50. This means that every 1 centimetre on the scale drawing represents 50 centimetres in real life. To be able to work out the measurements required on the scale drawing, you must first convert all measurements to centimetres. The base is 2.5 metres, which equals 250 centimetres. The shortest vertical side is 3 metres, which equals 300 centimetres. The longest vertical side is 3.75 metres, which equals 375 centimetres.

Now to work out the length of each side on the scale drawing. Since 1 centimetre on the drawing is 50 centimetres in real life, you need to divide the real-life measurement by 50 in order to work out the length required on the drawing. The base, 250, divided by 50, equals 5 centimetres. The shortest vertical side, 300, divided by 50, equals 6 centimetres. The longest vertical side, 375, divided by 50, equals 7.5 centimetres. To get the diagonal length, you simply join the tops of the two other sides together. Now work through the next few examples.

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Box _unit4.5.2 Summary of method

Remember if you are given the scale drawing measurement and asked to work out the real-life size, you need to multiply.

If you are given the real-life size and asked to work out the drawing measurement, you need to divide.

Case study _unit4.5.1 Example 1: Plan of a room

Rhodri has drawn a scale diagram of his living room. The scale he has used is 1:20.

On his diagram his living room is 30 cm long and 20 cm wide. What are the actual dimensions of his living room?

You need to work out the length and width separately.

Length

The scale he has used is 1:20, which means that everything is 20 times bigger in real life than on his diagram. You know the drawing measurement (30 cm) so you need to multiply by 20 to find the actual length of the room:

30 cm × 20 = 600 cm

It is a good idea to express the actual measurement in metres:

600 cm = 6 m

Width

You need to use the same scale of 1:20. If the width of the living room on the drawing is 20 cm then it will be 20 times as great in real life:

20 cm × 20 = 400 cm (400 cm = 4 m)

So Rhodri’s actual living room measures 6 m by 4 m.

Now have a go at solving these problems involving scale. Do the calculations without a calculator. You may double-check on a calculator if you need to and make sure you check your answers against ours.

Activity _unit4.5.2 Activity 10: Scale problems

A scale drawing has been drawn below of a shed that a garden planner wants to build. The scale used for the drawing is 1:25.
The area that the shed will be built on is a rectangle which measures 5.1 m by 4 m. Will the shed fit into the space allocated?
Figure _unit4.5.2 Figure 51 Calculating scale 1
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A scale drawing of the plot area of an L-shaped shed. The length of the vertical side is labelled ‘15.2 cm’. The length of the horizontal side is labelled ‘12.5 cm’.
Figure 51 Calculating scale 1

Answer

You know the scale is 1:25, so 1 cm on the diagram represents 25 cm in real life. You have been given the measurements on the diagram and want to work out the actual measurements so you need to multiply.
Horizontal length:
- 12.5 × 25 = 312.5 cm = 3.125 m
Vertical length:
- 15.2 × 25 = 380 cm = 3.8 m
Since both lengths for the shed are shorter than the lengths given for the area of land, you know the shed will fit.

A landscaper wants to put a wild area in your garden. She makes a scale plan of the wild area:
Figure _unit4.5.3 Figure 52 Calculating scale 2
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An illustration of a wild area for a garden showing bushes, flowers and butterflies. The bottom length is labelled ‘9 cm’ with ‘Scale 1:50’ underneath.
Figure 52 Calculating scale 2
What is the length of the longest side of the actual wild area in metres?

Answer

The length on the drawing is 9 cm, and the scale is 1:50. This means that 1 cm on the drawing is equal to 50 cm in real life. So to find out what 9 cm is in real life, you need to multiply it by 50:
- 9 × 50 = 450 cm
The question asks for the length in metres, so you need to convert centimetres into metres:
- 450 ÷ 100 = 4.5 m
The actual length of the wild area will be 4.5 m.

Here is a scale drawing showing one disabled parking space in a supermarket car park. The supermarket plans to add two more disabled parking spaces either side of the existing one.
Figure _unit4.5.4 Figure 53 Calculating scale 3
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An illustrated bird’s-eye view drawing of six car parking spaces with two rows of three. There is a disabled sign in the centre space in the bottom row. Under the disabled space is the width label of ‘2 cm’. To the left of the width label is ‘Scale: 1:125’. There is a red car in the top left space and bottom right space and a blue car in the top right space.
Figure 53 Calculating scale 3
What will be the total actual width of the three disabled parking spaces in metres?

Answer

You need to find out the width of three disabled parking spaces. The width of one parking space on the scale drawing is 2 cm, so first you need to multiply this by 3:
- 2 × 3 = 6 cm
The scale is 1:125. This means that 1 cm on the drawing is equal to 125 cm in real life. So to find out what 6 cm is in real life, you need to multiply it by 125:
- 6 × 125 = 750 cm
The question asks for the length in metres, so you need to convert centimetres into metres:
- 750 ÷ 100 = 7.5 m
The actual width of all three parking bays will be 7.5 m.

For this question you will need a pen or pencil, paper and a ruler.
Jane has a raised vegetable patch. She plans to build a slope leading up to the vegetable patch. Jane will cover the slope with grass turf.
She draws this sketch of the cross-section of the slope. The measurements indicated are the actual measurements.
Figure _unit4.5.5 Figure 54 Finding the length of a slope
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A sketch of a right-angled triangle. The vertical side (height) is labelled ‘45 cm’. The horizontal side (length) is labelled ‘125 cm’. The third side is labelled ‘length of slope’.
Figure 54 Finding the length of a slope
Jane will use a scale diagram to work out the length of the slope. She wants to use a scale of 1:10.
- a.What will the measurements of the base and height of the slope be on her diagram?
- b.Draw her scale diagram using the measurements you calculated in part (a).
  (i) What will the length of the slope be on the diagram?
  (ii) What will it be in real life?

Answer

- a.The scale is 1:10 and you want to work out the measurements for the scale diagram, so you need to divide the real-life measurements by 10:
  Base of patch = 125 ÷ 10 = 12.5 cm
  Height of patch = 45 ÷ 10 = 4.5 cm
- b. (i) Draw the scale diagram with the base measuring 12.5 cm and the height measuring 4.5 cm.
  Now draw in your slope linking the end points of the base and height together. If you measure the length of the slope with a ruler it should measure around 13.2 cm on the diagram.
  (ii) To work out the length of the actual slope, you need to use the scale of 1:10 again, but this time you need to multiply by 10 to work out the actual slope length:
  13.2 cm × 10 = 132 cm in real life.
  Your answer may vary slightly from ours but should be within a reasonable range.

Summary

In this section you have:

applied your ratio skills to the concept of scale plans and drawings
interpreted scale plans.

Well done! You have now completed this section of your course. You are now ready to test the knowledge and skills you’ve learned in the end of session quiz. Good luck!

6 Session 3 quiz

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Now it’s time to review your learning in the end-of-session quiz.

Session 3 quiz.

Open the quiz in a new window or tab (by holding ctrl [or cmd on a Mac] when you click the link), then return here when you have done it.

7 Session 3 summary

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Well done! You have now completed Session 3 ‘Shape and space’. If you have identified any areas that you need to work on, please ensure you refer back to this section of the course.

You should now be able to:

understand the difference between perimeter, area and volume and be able to calculate these for both simple and more complex shapes
know that volume is a measure of space inside a 3D object and calculate volumes of shapes in order to solve practical problems
draw and use a scale drawing or plan.

All of the skills above will help you with tasks in everyday life. Whether you are at home or at work, number skills are essential skills to have.

You are now ready to move on to Session 4, ‘Handling data’.

Introduction

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Data is a big part of our lives and can be represented in many different ways. This session will take you through a number of these representations and show you how to interpret data to find specific information. Before delving into the world of charts, graphs and averages though, it is important to make the distinction between the two different types of data:

Qualitative data – this is data that is not numerical e.g. eye – colours, favourite sports, models of cars.
Quantitative data – this is numerical data related to things that can be counted or measured e.g. temperatures, house prices, GCSE grades. Quantitative data can be discrete or continuous.

By the end of this session you will be able to:

identify different types of data
create and use tally charts, frequency tables and data collection sheets to record information
draw and interpret bar charts, pie charts and line graphs
understand there are different types of averages and be able to calculate each type
understand that probability is about how likely an event is to happen and the different ways that it can be expressed.

Download this video clip.Video player: bltl_a70_data.mp4

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Transcript

[MUSIC PLAYING]

We all use and generate data on a daily basis. For example, companies use data to track our shopping and offer us deals. Data has become very important.

There are many different types of data. Continuous data is something you measure, like, the length of a piece of wood. A box of nails is usually something you can count, which is called discrete data.

Different ways of displaying data include using graphs. A line graph can show the relationship between some data about a shelf. A pie chart can show the ways that someone spends their time, which is easy to see and understand as a picture.

Gathering data can help to work out different types of average – like the range, which is the difference between the lowest and highest values. Or the median, which is basically, the middle.

Data can even be used to work out probability – how likely are unlikely something is to happen, which can help you avoid unpleasant surprises.

[GLASS SHATTERING]

[LAUGHING]

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1 Discrete and continuous data

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Discrete data is information that can only take certain values. These values don’t have to be whole numbers (a child might have a shoe size of 3.5 or a company may make a profit of £3456.25 for example) but they are fixed values – a child cannot have a shoe size of 3.72!

The number of each type of treatment a salon needs to schedule for the week, the number of children attending a nursery each day or the profit a business makes each month are all examples of discrete data. This type of data is often represented using tally charts, bar charts or pie charts.

Continuous data is data that can take any value. Height, weight, temperature and length are all examples of continuous data. Some continuous data will change over time; the weight of a baby in its first year or the temperature in a room throughout the day. This data is best shown on a line graph as this type of graph can show how the data changes over a given period of time. Other continuous data, such as the heights of a group of children on one particular day, is often grouped into categories to make it easier to interpret.

You will have looked at different ways of presenting data in Everyday maths 1. Have a go at the next activity to see what you can remember.

Activity _unit5.1.1 Activity 1: Presenting discrete and continuous data

Match the best choice of graph for the data below.

Chart to show the heights of children in a class.
Chart to show favourite drink chosen by customers in a shopping centre.
Chart to show the temperature on each day of the week.
Chart to show percentage of each sale of ticket type at a concert.

Figure _unit5.1.1 Figure 1 Different types of charts and graphs

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Illustration of four types of chart and graph. (a) A pie chart; (b) a tally chart; (c) a line graph; (d) a bar chart.

Figure 1 Different types of charts and graphs

Answer

Chart to show the heights of children in a class.
The best choice here is (d) the bar chart as it can show each child’s height clearly.
Chart to show favourite drink chosen by customers in a shopping centre.
The best choice here is (b) the tally chart since you can add to this data as each customer makes their choice. A bar or pie chart would also be suitable.
Chart to show the temperature on each day of the week.
The only choice here is (c) the line graph as it shows how the temperature changes over time.
Chart to show percentage of each sale of ticket type at a concert.
The best choice here is probably (a) the pie chart since it shows clearly the breakdown of each type of ticket sale. A bar chart would also represent the data suitably.

Now that you are familiar with the two different types of data let’s look in more detail at the different types of chart and graph; how to draw them accurately and how to interpret them.

Summary

In this section you have:

learned about the two different types of data, discrete and continuous, and when and why they are used.

2 Tally charts, frequency tables and data collection sheets

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Have you ever been stopped in the street or whilst out shopping and asked about your choice of mobile phone company or to sample some food or drink and give your preference on which is your favourite? If you have, chances are, the person who was conducting the survey was using a tally chart to collect the data.

Figure _unit5.2.1 Figure 2 Favourite fruit tally chart

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Illustration showing a clipboard with a tally chart labelled ‘Favourite Fruit’ that has two columns labelled ‘Fruit’ and ‘Tally’. The fruit column rows are orange, apple, banana, strawberry and pineapple. The tally column rows show 11, 10, 8, 7 and 4 tally marks.

Figure 2 Favourite fruit tally chart

Tally charts are convenient for this type of survey because you can note down the data as you go. Once all the data has been collected it can be counted up easily because, as shown in the picture above, every fifth piece of data for a choice is marked as a diagonal line. This allows you to count up quickly in fives to get the total. A frequency or total column can then be filled out to make the data easier to work with.

Take a look at the example below:

Table _unit5.2.1 Table 1
Method of Travel	Tally	Frequency
Walk		9
Bike		3
Car		6
Bus		12
	TOTAL	30

You can see each category and its total, or frequency, clearly. Whilst this is a very simple example it demonstrates the purpose of a tally chart well. A tally chart may often be turned into a bar chart for a more visual representation of the data but they are useful for the actual data collection.

You can also use a tally chart for collecting grouped data. If for example, you want to survey the ages of clients or customers, you would not ask for each person’s individual age, you would ask them to record which age group they came within. If you want to set yourself up a tally chart for this data it might look similar to the below:

Table _unit5.2.2 Table 2
Age	Tally	Frequency
0–9
10–19
20–29
30–39
40–49
50–59
60–69
70+

Note that the age groups do not overlap; a common mistake would be to make the groups 0–10, 10–20 and so on. This is incorrect because if you were aged 10, you would not know which group you should place yourself in.

A more complex example of a tally chart can be seen below. In this example you can see that the information that has been collected is split into more than one category. These are sometimes called data summary sheets or data collection tables. It does not matter where each category is placed on the chart as long as all aspects are included.

Table _unit5.2.3 Table 3
	fewer than 6 trips		6 trips or more
	under 26 years	26 years and over	under 26 years	26 years and over
male	(1)	(0)	(3)	(6)
female	(3)	(0)	(1)	(4)

If you want to design a data collection or data summary sheet, you first need to know which categories of information you are looking for. Let’s take a look at an example of how you might do this.

Imagine you work in a hotel and want to gather some data on your guests. You want to know the following information:

Rating given by the guest: excellent, good, or poor.
Length of stay: under 5 days, or 5 days or more.
Location: from the UK, or from abroad.

A data summary sheet for this information could look like this:

Table _unit5.2.4 Table 4
	Stayed for under 5 days			Stayed for 5 days or more
	Excellent	Good	Poor	Excellent	Good	Poor
From the UK
From abroad

Have a go at the activity below and try creating a data collection sheet for yourself.

Activity _unit5.2.1 Activity 2: Data collection

You work at a community centre and want to gather some data on the people who use your services. You would like to know the following information:

whether they are male or female
if they use the centre during the day or during the evening
which age category they are in: 0–25, 26–50, 51+.

Design a suitable data collection sheet to gather the information.

Answer

Your chart should look something like the example below. You may have chosen to put the categories in different rows and columns, which is perfectly fine. As long as all the options are covered your data collection sheet will be correct.

Table _unit5.2.5 Table 5
	Daytime Visitor			Evening Visitor
	0–25	26–50	51+	0–25	26–50	51+
Male
Female

Now that you know the different ways in which you can collect your data, it’s time to look at how you can put this information into various charts and graphs.

Summary

In this section you have:

explored the differences between tally charts, frequency tables and data collection sheets and understood their usefulness in data collection.

3 Bar charts

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

There are two main types of bar charts (also known as bar graphs) – single and dual. The most common type is a single bar chart. These bar charts use vertical or horizontal bars to display data as seen below.

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Figure _unit5.3.1 Figure 3 Vertical and horizontal single bar charts

A comparative bar chart (also known as a dual bar chart) shows a comparison between two or more sets of data. Whilst the chart below shows the favourite sports of a group of students, it further breaks down the data and gives a comparison between boys and girls; there are two bars for each sport. Depending on the type of data you are collecting, you could have more than two bars under each category.

Looking at the example comparative bar chart below, we can see that 6 girls said their favourite sport was football compared to only 3 boys.

Figure _unit5.3.2 Figure 4 Dual bar chart

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Dual bar charts labelled ‘Our favourite sports’ with horizontal axes labelled ‘Sports’, vertical axes labelled ‘Number of students’ and bar categories labelled ‘Football’, ‘Hockey’, ‘Rugby’ and ‘Other’. The dual bar chart has two bars per category; one labelled ‘Boys’ (blue) and one labelled ‘Girls’ (orange).

Figure 4 Dual bar chart

A third way of presenting data is a component bar chart. Instead of using separate bars, all the data is represented in a single bar. Each component, or part, of the bar is coloured differently. A key is again needed to show what each colour represents. Look at the example below:

Figure _unit5.3.3 Figure 5 Single (component) bar chart

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Single bar chart labelled ‘Our favourite sports’ with horizontal axes labelled ‘Sports’, vertical axes labelled ‘Number of students’ and bar categories labelled ‘Football’, ‘Hockey’, ‘Rugby’ and ‘Other’. The single bar chart has one bar for each category and each bar is split horizontally for each category of ‘Boys’ (blue) and ‘Girls’ (orange).

Figure 5 Single (component) bar chart

A component bar chart is useful for displaying data where we want to see the overall total together with a breakdown. In the chart above, we can easily see that in total 9 students said that football was their favourite sport. However, we can also see the data broken down to show us that 3 boys and 6 girls said that football was their favourite sport.

3.1 Features of a bar chart

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Regardless of which type of bar chart you are drawing, there are certain features that all charts should have. Look at the diagram below to learn what they are.

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Figure _unit5.3.4 Figure 6 Features of a bar chart

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Illustration of the bar chart in the Figure 4. Labels point to features of the chart. The text reads: Key – bar charts with more than one bar for each section will need a key so it is clear what each bar refers to. Bars – whilst this seems like stating the obvious, your bar graph needs accurately drawn bars that are all the same width. Horizontal axis title – all bar charts need a title along the axis to describe what the information along the axis relates to. Numbered scale – the axis must be numbered with a sensible scale, the numbers could go up in 1s, 2s, 5s, 10s etc. depending on what is appropriate for the data. Vertical axis title – all bar charts need a title along this axis to describe what the information along this axis refers to.

Figure 6 Features of a bar chart

If you are drawing a graph or chart by hand, you will need to use graph paper. However, creating a graph on a computer is much quicker and more convenient than doing it by hand. The computer will select the scale for the graph and it will plot it accurately. All you need to do is make sure you choose the best type of graph for your data and remember to label it all clearly and appropriately.

Activity _unit5.3.1 Activity 3: Drawing a bar chart

Draw a bar chart to represent the information shown in the table below. You may hand-draw your graph or create it using a computer package.

Table _unit5.3.1 Table 6 Leisure centre users
	January	February	March	April
Male	175	154	120	165
Female	205	178	135	148

Remember to choose a suitable scale for your chart and ensure that it has a title and axis labels. It will also need a key.

Answer

Your bar chart should look something like the example below. You may have chosen to use a slightly different numbering scale but your chart should have the following features:
- a title
- labels on both the horizontal and vertical axes
- bars labelled with the month
- accurately drawn bars of the same width
- a numbered scale on the vertical axis
- a key to show which bars are for males and which are for females.
Figure _unit5.3.5 Figure 7 A bar chart showing numbers of leisure centre users
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A bar chart showing the information provided in Table 6.
Figure 7 A bar chart showing numbers of leisure centre users

Draw a bar chart to represent the data shown in the table below. The numbers are much bigger than in the previous example so if you are drawing your chart by hand, you will need to consider your scale carefully.

Table _unit5.3.2 Table 7 **Food sales at a three-day summer music festival**
Food types	Number of portions sold
Sandwiches and baguettes	37 000
Burgers	25 000
Fish and chips	16 000
Ice cream cones	12 000
Other	8 500

Answer

Your bar chart should look something like the example below.
Note the use of gridlines to help with interpreting the data in the graph. Also note how the scale and label on the vertical axes (see the second example on the right-hand side of the graph) are slightly different. Both formats are shown here as different examples and either would be acceptable.
Figure _unit5.3.6 Figure 8 A bar chart showing food sales at three-day music festival
Show description|Hide description
A bar chart showing the information provided in Table 7. A second vertical axis is shown on the right of the graph. The left vertical axis says: ‘Number of portions sold’ with the scale showing numbered intervals at 10 000, 20 000, 30 000 and 40 000. The right vertical axis says: ‘Number of portions sold (000s)’ with the scale showing numbered intervals at 10, 20, 30 and 40.
Figure 8 A bar chart showing food sales at three-day music festival
Your bar chart should be similar to the above and include:
- a title
- labels on both the horizontal and vertical axes
- bars labelled with the food types
- accurately drawn bars of the same width
- a numbered scale on the vertical axis, which goes up in equal intervals.

Now that you understand the features of bar charts and are able to draw them, let’s look at how to interpret the information they show.

3.2 Interpreting bar charts

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

If you see a bar chart that is showing you information, it’s useful to know how to interpret the information given. The most important thing you need to read to understand a bar chart correctly is the scale on the vertical axis.

As you know from drawing bar charts yourself, the numbers on the vertical axis can go up in steps of any size. You therefore need to work out what each of the smaller divisions is worth before you can use the information. The best way for you to learn is to practise looking at, and reading from, different bar charts.

Have a go at the activity below and see how you get on.

Activity _unit5.3.2 Activity 4: Interpreting a bar chart

The chart below shows the average daily viewing figures for Sky Cinema channels during 11–17 March 2019. Answer the questions that follow. Choose the correct answer in each case.

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Figure _unit5.3.7 Figure 9 Sky Cinema average viewing figures (11–17 March 2019)

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The bar chart shows the following viewing figures for Sky Cinema during 11–17 March 2019:

Action & Adventure – 264 000

Comedy – 150 000

Crime & Thriller – 111 000

Disney – 254 000

Drama & Romance – 72 000

Family – 235 000

Greats – 126 000

SciFi-Horror – 172 000

Figure 9 Sky Cinema average viewing figures (11–17 March 2019)

Which was the most popular channel?
- a.Action & Adventure
- b.Disney
- c.Family

Answer

a.Action & Adventure

Approximately how many viewers tuned in to Sky Cinema Disney?
- a.250 viewers
- b.25 000 viewers
- c.250 000 viewers

Answer

c.250 000 viewers – note how the vertical axis label states that the figures are in 000s (thousands) which means that the scale goes up in intervals of 20 000. This means the Disney bar goes up to 250 000.

Approximately how many more viewers watched Sky Cinema SciFi-Horror than Sky Cinema Drama & Romance?
- a.100 000
- b.80 000
- c.100

Answer

a.100 000 – Sky Cinema SciFi-Horror had appoximately 175 000 viewers whilst Sky Cinema Drama & Romance had approximately 75 000 viewers, making a difference of 100 000 viewers.

Activity _unit5.3.3 Activity 5: Interpreting a comparative bar chart

The chart below shows the average house prices by local authority in Wales for 2017 and 2018. Answer the questions that follow.

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Figure _unit5.3.8 Figure 10 Average house price by local authority in January

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The bar chart shows the following average house price by local authority in 2017 (blue bar) and 2018 (orange bar):

Cardiff – £200 082 (orange), £190 244 (blue)

Ceredigion – £179 419 (orange), £170 682 (blue)

Flintshire – £167 673 (orange), £158 629 (blue)

Gwynedd – £148 891 (orange), £145 362 (blue)

Isle of Anglesey – £178 247 (orange), £157 541 (blue)

Pembrokeshire – £166 374 (orange), £163 386 (blue)

Powys – £179 789 (orange), £172 568 (blue)

Swansea – £142 592 (orange), £138 422 (blue)

Figure 10 Average house price by local authority in January

Which local authority had the biggest average house price increase between 2017 and 2018?

Answer

Isle of Anglesey

Which of the following is the best estimate of the difference between Cardiff and Swansea’s 2018 average house prices?
- a.£50 000
- b.£60 000
- c.£120 000

Answer

b.£60 000 – note how the scale goes up in intervals of £10 000. In 2018 Cardiff’s average house price was approximately £200 000 compared to Swansea’s average house price of approximately £140 000, making a difference of £60 000.

Activity _unit5.3.4 Activity 6: Interpreting a component bar chart

The following chart shows the number of enrolments for different college courses. Answer the questions that follow.

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Figure _unit5.3.9 Figure 11 College course enrolments

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A bar chart of college course enrolments showing the following data:

Painting & Decorating = 20

Health & Social Care = 40

Travel & Tourism = 12

Business Admin = 36

Carpentry & Joinery = 24

Hair & Beauty = 65

Figure 11 College course enrolments

Which course had the fewest number of part-time enrolments?

Answer

Travel & Tourism

Which course had more part-time than full-time enrolments?

Answer

Health & Social Care

How many full-time enrolments did Hair & Beauty have?

Answer

49.
Note how the scale goes up in 2s and the full-time component of the Hair & Beauty bar goes up to halfway between 48 and 50.

How many part-time enrolments did Business Admin have?

Answer

Approximately how many course enrolments were there in total?

Answer

197 – You need to count the total enrolments for all of the separate courses and then add them together:
- Painting & Decorating = 20
- Health & Social Care = 40
- Travel & Tourism = 12
- Business Admin = 36
- Carpentry & Joinery = 24
- Hair & Beauty = 65
20 + 40 + 12 + 36 + 24 + 65 = 197

You should now be feeling confident with both drawing and interpreting bar charts so it’s time to move on to look at another type of chart – pie charts.

Summary

In this section you have learned:

about the different features of a bar chart
to use single and comparative bar charts to present data
to interpret the information shown on different types of bar chart.

4 Pie charts

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

A pie chart is the best way of showing the proportion (or fraction) of the data that is in each category. It is very easy to visually see the largest and smallest sections of the chart and how they compare to the other sections.

This section will help you to understand when a pie chart should be used over other presentation methods. In the first part of this section you’ll learn how to draw and then move on to interpreting pie charts.

Figure _unit5.4.1 Figure 12 A partially eaten pie

In the first part of this section you’ll learn how to draw and then move on to interpreting pie charts.

4.1 Drawing pie charts

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

The best way to understand the steps involved in drawing a pie chart is to watch the worked example in the video below.

Download this video clip.Video player: bltl_3_4_1_pie_charts.mp4

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Transcript

When you draw a pie chart, you're splitting a circle into a number of different sections of different sizes, depending on the data you're given. A full circle has a total of 360 degrees. A quarter of a circle is 90 degrees, and a half circle is 180 degrees.

Let's create a pie chart to show you the steps involved in drawing one. This table shows some information about the number of clients at a beauty salon and what treatments they had. If you want to draw a pie chart, you need to know how many degrees of the circle to allocate for each treatment.

To do this, you need to divide 360 degrees by the number of clients. This is because you want to share the full circle, 360 degrees, equally between each of the clients. Each person is represented by a certain number of degrees.

In this example, the total number of clients equals 24 + 16 + 15 + 5, which equals 60. 360 divided by 60 equals 6, so each client gets six degrees of the circle.

Now that you know how many degrees each treatment type is worth, you can draw the pie chart. First, you need to draw a circle. Next, draw a line from the centre of the circle to the top of the circle. This will be the line that you start drawing your slices from. Now, use a protractor to measure the correct angles and draw the pie chart.

It should look something like this. Now, have a go at drawing a pie chart for yourself.

End transcript

Download

Interactive feature not available in single page view (see it in standard view).

Now have a go at drawing a pie chart for yourself.

Activity _unit5.4.1 Activity 7: Drawing a pie chart

A leisure centre wants to compare which activities customers choose to do when they visit the centre. The information is shown in the table below. Draw an accurate pie chart to show this information. You may hand-draw your pie chart or create it using a computer package.

Table _unit5.4.1 Table 8(a)
Activity	Number of customers
Swimming	26
Gym	17
Exercise class	20
Sauna	9

Answer

Firstly, work out the total number of customers:
- 26 + 17 + 20 + 9 = 72
Now work out the number of degrees that represents customer:
- 360˚ ÷ 72 = 5˚ per customer

Table _unit5.4.2 Table 8(b)
Activity	Number of customers	Number of degrees
Swimming	26	26 × 5 = 130˚
Gym	17	17 × 5 = 85˚
Exercise class	20	20 × 5 = 100˚
Sauna	9	9 × 5 = 45˚

Now use this information to draw your pie chart. It should look something like this:

Figure _unit5.4.2 Figure 13 Pie chart for customer leisure centre activities

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A pie chart labelled ‘Customer activities’ with each segment representing the data shown in Table 8(b).

Figure 13 Pie chart for customer leisure centre activities

The table below shows the sandwich sales over one year for sandwich company, Belinda’s Butties. Draw a pie chart to illustrate the data. You may hand-draw your pie chart or create it using a computer package.

Table _unit5.4.3 Table 9(a)
Sandwich Type	Sales (000s)
Cheese and onion	20
Egg and cress	17
Prawn	11
Coronation chicken	12
Tuna mayonnaise	9
Ham	8
Beef and tomato	13

Answer

Firstly, work out the total number of sandwich sales:
- 20 + 17 + 11 + 12 + 9 + 8 + 13 = 90 (000s)
Now work out the number of degrees that represents each sale:
- 360 ÷ 90 = 4˚ per sale

Table _unit5.4.4 Table 9(b)
Sandwich Type	Sales (000s)	Number of Degrees
Cheese and onion	20	20 × 4 = 80°
Egg and cress	17	17 × 4 = 68°
Prawn	11	11 × 4 = 44°
Coronation chicken	12	12 × 4 = 48°
Tuna mayonnaise	9	9 × 4 = 36°
Ham	8	8 × 4 = 32°
Beef and tomato	13	13 × 4 = 52°

Now use this information to draw your pie chart. It should look something like the one below.

Figure _unit5.4.3 Figure 14 Belindas Butties – sales over one year (000s)

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A pie chart with each segment representing the data shown in Table 9(b).

Figure 14 Belindas Butties – sales over one year (000s)

Now that you can accurately draw a pie chart, it’s time to look at how to interpret them. You won’t always be given the actual data, you may just be given the total number represented by the chart or a section of the chart and the angles on the pie chart itself. It’s useful to know how to use your maths skills to work out the actual figures.

Here’s a reminder of the degrees of a circle which will be useful when you come to read from pie charts.

Figure _unit5.4.4 Figure 15 Degrees of a circle

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Four circles with arrows around the outside showing 45°, 90°, 180° and 360°.

Figure 15 Degrees of a circle

4.2 Interpreting pie charts

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Imagine you’ve been presented with the pie chart below. The chart shows the ages of students competing at an athletic event.

Figure _unit5.4.5 Figure 16 Ages of students competing at an athletic event

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A pie chart with four segments labelled 13 years, 16 years, 15 years and 14 years. The 13 years segment has a square drawn in the corner labelled ‘Remember, an angle shown like this is a right angle, meaning it’s 90°. The 16 years segment is marked as 60° and the 14 years segment is marked as 115°.

Figure 16 Ages of students competing at an athletic event

There are two possible pieces of information you could be given. You could be given the total number of students that were at the event or, you could be given the number of students in one of the age categories.

Case study _unit5.4.1 Example: Reading a pie chart 1

Let’s say you were told that 72 people attended the competition. Since you know that 360˚ has been shared equally between all 72 people, you do 360 ÷ 72 = 5˚ per person.

Once you know this, if you wanted to know how many students that took part were 16 years old, you would look at the degrees on the chart for 16-year-olds which in this example is 60˚.

You then do 60 ÷ 5 = 12 students.

If you wanted to work out the number of 15-year-olds, you would first need to work out the missing angle; you know that all the angles will add up to 360˚ so just do:

Extract _unit5.4.1

360 – 115 – 90 – 60 = 95˚

And now do the same as before 95 ÷ 5 = 19 students who were 15 years old.

Case study _unit5.4.2 Example: Reading a pie chart 2

Using the same pie chart, let’s say that all you were told was that 23 students took part who were 14 years old.

You can see that the angle for 14-year-olds is 115˚ and you’ve been told that this represents 23 students.

To find out how many degrees each student gets, you do 115 ÷ 23 = 5˚ per student.

Once you know this you can find out how many students are represented by each other section in the same way as we did in example 1. For example, the 13-year-olds have an angle of 90˚.

To find out how many there are you do 90 ÷ 5 = 18 students who were 13 years old.

Pie charts are very similar to ratio. In ratio questions you are always looking to find out how much 1 part is worth; in pie chart questions you are looking to find how many degrees represents 1 person (or whatever object the pie chart is representing).

As well as being closely linked with ratio, pie charts also involve the use of your fractions skills. If, for example, you were asked what fraction of the students were 16 years old, you can show this as since the 16-year-olds are 60 degrees out of the total 360 degrees.

Using your fractions skills however, the fraction can be simplified to .

It’s time for you to practise your skills at interpreting pie charts. Have a go at the activity below and then check your answers with the feedback given.

Activity _unit5.4.2 Activity 8: Interpreting pie charts

The pie chart below shows how long a gardener spent doing various activities over a month.
Figure _unit5.4.6 Figure 17 Time spent doing gardening activities
Show description|Hide description
A pie chart with four segments labelled ‘weeding’, ‘digging’, ‘planting’ and ‘cutting the grass’. The digging segment is marked as 100°, the planting segment is marked as 80°, the cutting the grass segment is marked as 40° and the weeding segment is blank.
Figure 17 Time spent doing gardening activities
- a.What fraction of the time was spent planting? Give your answer in its simplest form.
- b.5 hours were spent digging. How long was spent on cutting the grass?

Answer

- a.Planting = = in its simplest form.
- b.Digging is 100˚ and you know that that was 5 hours.
  100˚ ÷ 5 = 20˚ for each hour.
  Since cutting the grass has an angle of 40˚, you do 40 ÷ 20 = 2 hours cutting the grass.

120 adults participating in an online course were asked if they felt there were enough activities for them to complete throughout the course.
The pie chart below shows the results.
Figure _unit5.4.7 Figure 18 Pie chart of opinions about an online course
Show description|Hide description
A pie chart with four segments labelled ‘too little’, ‘do not know’, ‘too much’ and ‘enough’. The too little segment is marked as 45°, the do not know segment is marked as 60°, the too much segment is marked as 105° and the enough segment is blank.
Figure 18 Pie chart of opinions about an online course
- a.What fraction of the adults thought there were too many activities? Give your answer in its simplest form.
- b.How many adults thought there were enough activities?

Answer

- a.Too much = = in its simplest form.
- b.You know that 120 adults took part in the survey. To find out how many degrees represents each adult, you do 360 ÷ 120 = 3 degrees per person.
  Next you need to know the angle for those who said there were enough activities. For this, you do:
  360 – 105 – 60 – 45 = 150˚
  Now you know this, you can do:
  150 ÷ 3 = 50 adults thought there were enough activities.

Well done! You can now draw and interpret bar charts and pie charts; both of which are good ways to represent discrete data. In the next part of this session, you will learn how to draw and interpret line graphs.

Summary

In this section you have learned:

what types of information can be represented effectively on a pie chart
how to use and interpret a pie chart
how to draw an accurate pie chart when given a set of data.

5 Line graphs

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Line graphs are a very useful way to spot patterns or trends over time. You will have looked at how to plot and interpret single line graphs from single data sources in Everyday maths 1. Now you will be looking at line graphs that show the results of two data sources.

The example below shows the monthly foreign exchange rate of £1 against the US dollar and the euro. You can clearly see how the pound was dropping in value against the euro and US dollar up to October 2016. Now that you understand how line graphs can be used and why they are useful, next you’ll learn how to draw and interpret them.

Figure _unit5.5.1 Figure 19 Monthly foreign exchange rate of £1 against the US dollar and the euro from May 2016 to January 2018.

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A line graph. The horizontal axis is labelled ‘Year’ and has 11 data points from ‘May-16’ to ‘Jan-18’. The vertical axis is labelled ‘Units of currency’ and has 11 data points from 1.0 to 1.50.

Euro data points are:

31 May 16 – 1.2846

31 Jul 16 – 1.1884

30 Sep 16 – 1.1722

30 Nov 16 – 1.1533

31 Jan 17 – 1.1613

31 Mar 17 – 1.1548

31 May 17 – 1.1696

31 Jul 17 – 1.1281

30 Sep 17 – 1.1186

30 Nov 17 – 1.1259

31 Jan 18 – 1.1331

US Dollar data points are:

31 May 16 – 1.4518

31 Jul 16 – 1.3141

30 Sep 16 1.3142

30 Nov 16 – 1.2431

31 Jan 17 – 1.2351

31 Mar 17 – 1.2348

31 May 17 – 1.2933

31 Jul 17 – 1.2994

30 Sep 17 – 1.3324

30 Nov 17 – 1.3219

31 Jan 18 – 1.3832

Figure 19 Monthly foreign exchange rate of £1 against the US dollar and the euro from May 2016 to January 2018.

Now that you understand how useful line graphs can be and how they can be used, next you’ll learn how to draw and interpret them.

5.1 Drawing line graphs

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Drawing a line graph is very similar to drawing a bar chart, and they have many of the same features.

Line graphs need:

a title
a label for the vertical axis (e.g. units of currency)
a numbered scale on the vertical axis
a label on the horizontal axis (e.g. month) so that it is clear to the reader what they are looking at.

The main difference when drawing a line graph rather than a bar chart, is that rather than a bar, you put a dot or a small cross to represent each piece of information. You then join each dot together with a line. There is significant debate over whether the dots should be joined with a curve or with straight line. Whilst the issue is (believe it or not!) hotly contested, the general consensus seems to be that dots should be joined with straight lines.

Activity _unit5.5.1 Activity 9: Drawing a line graph

Have a go at drawing a line graph to represent the data below.
The table below shows the temperatures in Tenby during the first two weeks of July 2018.

Table _unit5.5.1 Table 10
Date	Temperature High ˚C	Temperature Low ˚C
01/07/2018	25	15
02/07/2018	28	17
03/07/2018	29	12
04/07/2018	24	15
05/07/2018	26	13
06/07/2018	22	12
07/07/2018	23	12
08/07/2018	27	12
09/07/2018	26	14
10/07/2018	24	14
11/07/2018	22	14
12/07/2018	22	7
13/07/2018	24	12
14/07/2018	25	20

Answer

Your line graph should look similar to the one shown below with a title, key and axis labels.
Figure _unit5.5.2 Figure 20 A line graph of temperatures in Tenby, Wales in July 2018
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A line graph with the title ‘Temperatures in Tenby, Wales in July 2018’ showing the data in Table 10.
Figure 20 A line graph of temperatures in Tenby, Wales in July 2018

A clothing store has outlets in Llandudno and Aberystwyth. Use the data in the table below to draw a line graph comparing monthly sales between January and June.

Table _unit5.5.2 Table 11
Month		Jan	Feb	Mar	Apr	May	Jun
Sales (£000s)	Llandudno	29	15	19	20	23	24
Sales (£000s)	Aberystwyth	25	10	16	16	21	26

Answer

Your line graph should look similar to the one shown below with a title, key and axis labels.
Figure _unit5.5.3 Figure 21 A line graph showing half-year sales for Llandudno and Aberystwyth
Show description|Hide description
A line graph with the title ‘Half-year sales for Llandudno and Aberystwyth’ showing the data in Table 11.
Figure 21 A line graph showing half-year sales for Llandudno and Aberystwyth

5.2 Interpreting line graphs

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Interpreting line graphs is also very similar to the way you interpret bar charts; it’s just about using the scales shown on the graph to find out information.

Given that you’ve already learned and practised interpreting bar charts, you can jump straight into an activity on interpreting line graphs.

Activity _unit5.5.2 Activity 10: A population line graph

The graph shows information about the population of a village in thousands over a period of time.
Figure _unit5.5.4 Figure 22 Line graph showing village population over time
Show description|Hide description
A line graph titled ‘Village population’. The horizontal axis is labelled ‘Time’ and shows the years 1981, 1991, 2001 and 2011. The vertical axis is labelled ‘Population (1000s)’ and is numbered from zero to 12. The data points are: 1981 (6), 1991 (8), 2001 (7) and 2011 (10).
Figure 22 Line graph showing village population over time
- a.What was the population of the village in 1991?
- b.What was the increase in population from 1981 to 2011?
- c.By how much did the population drop between 1991 and 2001?

Answer

- a.In 1991 the population was 8000.
- b.In 1981 the population was 6000, in 2011 the population was 10 000. This is an increase of 10 000 − 6000 = 4000.
- c.In 1991 the population was 8000 and by 2001 it was 7000. 8000 − 7000 = 1000. So the population dropped by 1000.

Jen’s Gym kept a record of member numbers during 2018.
The graph below shows information about the results.
View larger image
Figure _unit5.5.5 Figure 23 A line graph showing gym membership numbers for 2018
Show description|Hide description
A line graph titled ‘Gym membership for 2018’. The horizontal axis is labelled ‘Month’ and shows the twelve months of the year. The vertical axis is labelled ‘Number of members’ and has five numbered points every 25 from 0 to 100.
The approximate data points on the graph for females (red line) are: January (99), February (95), March (85), April (82), May (80), June (75), July (70), August (68), September (65), October (65), November (74), December (67).
The approximate data points on the graph for males (blue line) are: January (94), February (99), March (93), April (90), May (85), June (81), July (77), August (74), September (74), October (78), November (75), December (69).
Figure 23 A line graph showing gym membership numbers for 2018
- a.Which was the only month when there were more female members than male members?
- b.Estimate the difference in membership numbers for October.
- c.In what month was the difference between male and female members smallest?

Answer

- a.January
- b.13 (+/− 1) allowed as this question is estimation
- c.November

How did you get on? Hopefully you were able to answer all the questions without too much difficulty. As long as you have worked out the scale correctly and read the question carefully, there’s nothing too tricky involved.

You have now covered each drawing and interpreted each different type of chart and graph so it’s time to move on to look at other uses for data: averages and range.

Summary

In this section you have learned:

which types of data can be suitably represented by a line graph and which are best suited to other types of charts.
how to interpret the information shown on a line graph
how to draw an accurate line graph for a given set of data.

6 Mean, median, mode and range

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

The three types of averages that you will be focussing on in this part of the section are mean, median and mode. You will also be looking at range. For Level 2 Essential Skills Wales, you need to be able to calculate each of these without using a calculator.

6.1 Range

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Figure _unit5.6.1 Figure 24 A mountain range of different sized peaks

Much like this stunning mountain range is made up of a variety of different sized mountains, a set of numerical data will include a range of values from smallest to biggest. The range is simply the difference between the biggest value and the smallest value. It shows how spread out a set of data is and can be useful to know because data sets with a big difference between the highest and lowest values can imply a certain amount of risk.

Let’s say there are two basketball players and you are trying to choose which player to put on for the last quarter. If one player has a large range of points scored per game (sometimes they score a lot of points but other times they score very few – meaning their scoring is variable) and the other player has a smaller range (meaning they are more consistent with their point scoring) it might be safest to choose the more consistent player.

Take a look at the example below.

A farmer takes down information about the weight, in kg, of apples that one worker collected each day on his apple farm.

Table _unit5.6.1 Table 12
Monday	Tuesday	Wednesday	Thursday	Friday	Saturday	Sunday
56 kg	70 kg	45 kg	82 kg	67 kg	44 kg	72 kg

In order to find the range of this data, you simply find the biggest value (82 kg) and the smallest value (44 kg) and find the difference:

82 – 44 = 38 kg

The range is therefore 38 kg.

Now let’s compare this worker to another worker whose information is shown in the table below.

Table _unit5.6.2 Table 13
Monday	Tuesday	Wednesday	Thursday	Friday	Saturday	Sunday
56 kg	60 kg	58 kg	62 kg	65 kg	49 kg	58 kg

This worker has a highest value of 65 kg and a lowest value of 49 kg. The range for this worker is therefore 65 – 49 = 16 kg. The second worker has a lower range than the first worker and is therefore a more consistent apple picker than the first worker, who is a more variable picker.

Now try one for yourself.

Activity _unit5.6.1 Activity 11: Finding the range

The table below shows the sales made by a café on each day of the week:

Table _unit5.6.3 Table 14
Monday	Tuesday	Wednesday	Thursday	Friday	Saturday	Sunday
£156.72	£230.54	£203.87	£179.43	£188.41	£254.70	£221.75

What is the range of sales for the café over the week?

Answer

Simply find the highest value: £254.70, and lowest value: £156.72, then find the difference:
- £254.70 − £156.72 = £97.98

A bowling team want to compare the scores for their players. The table below shows their results.

Table _unit5.6.4 Table 15
Name	Andy	Bilal	Caz	Dom	Ede
Highest score	176	175	162	170	150
Lowest score	148	145	142	165	116

Which player is the most consistent? Give a reason for your answer.

Answer

You need to look at the range for each player:
- Andy: 176 − 148 = 28
- Bilal: 175 − 145 = 30
- Caz: 162 − 142 = 20
- Dom: 170 − 165 = 5
- Ede: 150 − 116 = 34

The player with the smallest range is Dom and so Dom is the most consistent player.

Outside temperatures at a garden centre were taken daily over four weeks in January and displayed in the following table.

Table _unit5.6.5 Table 16 Temperatures for January in ˚C
	Mon	Tue	Wed	Thu	Fri	Sat	Sun
Week 1	2	4	5	5	1	−1	−3
Week 2	−4	0	0	3	6	5	6
Week 3	3	2	−1	0	3	2	0
Week 4	0	4	7	8	3	−1	−2

- a.Which day of the week showed the most variable temperature range?
- b.Which day of the week showed the most consistent temperature range?
- c.Which days had the same range?

Answer

- a.Sundays had a lowest temperature of −3˚C and a highest temperature of 6˚C so the difference was 9˚C, giving Sundays the highest range and making temperatures more variable.
- b.Tuesdays had a lowest temperature of 0˚C and a highest temperature of 4˚C so the difference was 4˚C, giving Tuesdays the lowest range and making temperatures more consistent.
- c.Wednesdays and Thursdays had the same range.
  Wednesdays had a lowest temperature of −1˚C and a highest temperature of 7˚C so the difference was 8˚C.
  Thursdays had a lowest temperature of 0˚C and a highest temperature of 8˚C so the difference was 8˚C.

As you have seen, finding the range of a set of data is very simple but it can give some useful insights into the data. The most commonly used average is the ‘mean average’ (or sometimes just the mean) and you’ll look at this next.

6.2 Mean average

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

You will now learn about:

finding the mean when given a set of data
finding the mean from a frequency table

Figure _unit5.6.2 Figure 25 Below average and mean average

The mean is a good method to use when you want to compare a large set of data, for example:

the average amount spent by customers in a shop
the average cost of a house in a certain area
the average time taken for your chosen breakdown service to get to your car.

The mean average can help us make comparisons between sets of data which can then help you when making a decision.

6.3 Finding the mean from a set of data

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

To find the mean of a simple set of data, all you need to do is find the total, or sum, of all the items together and then divide this total by how many items of data there are.

Table _unit5.6.6 Table 13 (repeated)
Monday	Tuesday	Wednesday	Thursday	Friday	Saturday	Sunday
56 kg	60 kg	58 kg	62 kg	65 kg	49 kg	58 kg

Look again at the weight, in kg, of apples that one worker collected each day on an apple farm (shown above). If you want to calculate the mean average weight of apples collected, you first find the sum or total of the weight of apples collected in the week:

56 + 60 + 58 + 62 + 65 + 49 + 58 = 408 kg

Next, divide this total but the number of data items, in this case, 7:

408 ÷ 7 = 58.3 kg (rounded to one d.p.)

It is important to note that the mean may well be a decimal number even if the numbers you added together were whole numbers.

Another important thing to note here is that the two sums (the addition and then the division) are done as two separate sums. If you were to write:

56 + 60 + 58 + 62 + 65 + 49 + 58 ÷ 7

this would be incorrect (remember BIDMAS from Session 1?). Unless you are going to use brackets to show which sum needs to be done first (56 + 60 + 58 + 62 + 65 + 49 + 58) ÷ 7, it is accurate to write two separate calculations.

Have a go at calculating the mean for yourself by completing the activity below.

Activity _unit5.6.2 Activity 12: Finding the mean

The table below shows the sale price of ten, 2 bedroom semi-detached houses in a town in Liverpool.

Table _unit5.6.7 Table 16
1	2	3	4	5	6	7	8	9	10
£70 000	£65 950	£66 500	£71 200	£68 000	£62 995	£70 500	£68 750	£59 950	£67 900

What is the mean house price in this area?

Answer

First find the total of the house prices:
- £70 000 + £65 950 + £66 500 + £71 200 + £68 000 + £62 995 + £70 500 + £68 750 + £59 950 + £67 900 = £671 745
Now divide this total by the number of houses (10):
- £671 745 ÷ 10 = £67 174.50

The table below shows the units of gas used by a household for the first 6 months of a year.

Table _unit5.6.8 Table 17
January	February	March	April	May	June
1650	1875	1548	1206	654	234

Calculate the mean amount of gas units used per month.

Answer

Find the total number of units used:
- 1650 + 1875 + 1548 + 1206 + 654 + 234 = 7167 units
Now divide this total by the number of months (6):
- 7167 ÷ 6 = 1194.5 units

This method of finding the mean is fine if you have a relatively small set of data. What about if the set of data you have is much larger? If this was the case, the data would probably not be presented as a list of numbers, it’s much more likely to be presented in a frequency table.

In the next part of this section, you will learn how find the mean when data is presented in this way.

6.4 Finding the mean from a frequency table

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Large groups of data will often be shown as a frequency table, rather than as a long list. This is a much more user friendly way to look at a large set of data. Look at the example below where there is data on how many times, over a year, customers used a gardening service.

Table _unit5.6.9 Table 18
Number of visits in a year	Number of customers
1	6
2	10
3	11
4	16
5	4
6	3

We could write this as a list if we wanted to:

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 and so on

but it’s much clearer to look at in the table format. But how would you find the mean of this data? Well, there were 6 customers who had 1 visit from the gardening company, that’s a total of 1 × 6 = 6 visits. Then there were 10 customers who had 2 visits, that’s a total of 2 × 10 = 20 visits. Do this for each row of the table, as shown below.

Table _unit5.6.10 Table 19
Number of visits in a year	Number of customers	Total visits
1	6	1 × 6 = 6
2	10	2 × 10 = 20
3	11	3 × 11 = 33
4	16	4 × 16 = 64
5	4	5 × 4 = 20
6	3	6 × 3 = 18
	Total = 50 customers	Total = 161 visits

Finally, work out the totals for each column (highlighted in a lighter colour on the table above).

Now you have all the information you need to find the mean; the total number of visits is 161 and the total number of customers is 50 so you do 161 ÷ 50 = 3.22 visits per year as the mean average.

Warning! Many people trip up on these because they will find the total visits (161) but rather than divide by the total number of customers (50) they divide by the number of rows in the table (in this example, 6).

If you do 161 ÷ 6 = 26.83, logic tells you, that since the maximum number of visits any customer had was 6, the mean average cannot be 26.83. Always sense check your answer to make sure it is somewhere between the lowest and highest values of the table. For this example, anything below 1 or above 6 must be incorrect!

Have a go at a couple of these yourself so that you feel confident with this skill.

Activity _unit5.6.3 Activity 13: Finding the mean from frequency tables

The table below shows some data about the number of times children were absent from school over a term.
Work out the mean average number of absences.

Table _unit5.6.11 Table 20(a)
Number of absences	Number of children
1	26
2	13
3	0
4	35
5	6

Answer

First, work out the total number of absences by multiplying the absence column by the number of children. Next, work out the totals for each column.

Table _unit5.6.12 Table 20(b)
Number of absences	Number of children	Total number of absences
1	26	1 × 26 = 26
2	13	2 × 13 = 26
3	0	3 × 0 = 0
4	35	4 × 35 = 140
5	6	5 × 6 = 30
	Total = 80	Total = 222

To find the mean do: 222 ÷ 80 = 2.775 mean average.

You run a youth club for 16–21-year-olds and are interested in the average age of young people who attend.
You collect the information shown in the table below. Work out the mean age of those who attend. Give your answer rounded to one decimal place.

Table _unit5.6.13 Table 21(a)
Age	Number of young people
16	3
17	8
18	5
19	12
20	6
21	1

Answer

Again, first work out the totals for each row and column.

Table _unit5.6.14 Table 21(b)
Age	Number of young people	Total number
16	3	16 × 3 = 48
17	8	17 × 8 = 136
18	5	18 × 5 = 90
19	12	19 × 12 = 228
20	6	20 × 6 = 120
21	1	21 × 1 = 21
	Total = 35	Total = 643

To find the mean you then do:
- 643 ÷ 35 = 18.4 years (rounded to one d.p.)

If you are given all the data in a set and asked to find the mean, it’s a relatively simple process.

For the Confirmatory Test element of Essential Skills Wales (ESW), you are expected to complete all of the necessary calculations without using a calculator, so remember to use alternative methods for checking e.g. inverse checks.

6.5 Calculating the median

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

The next type of average you look at briefly is called the median. Put very simply, the median is the middle number in a set of data and as it is in the middle, it is not affected by abnormally high or low data values. The important thing you need to remember is to put the numbers in size order, smallest to largest, before you begin. Firstly let’s look at two simple examples.

Case study _unit5.6.1 Example: Finding the median 1

Find the median of this data set:

Method

5, 10, 8, 12, 4, 7, 10

Firstly, order the numbers from smallest to largest:

4, 5, 7, 8, 10, 10, 12

Now, find the number that is in the middle:

4, 5, 7, 8, 10, 10, 12

8 is the number in the middle, so the median is 8.

Case study _unit5.6.2 Example: Finding the median 2

Find the median of this data set:

Method

24, 30, 28, 40, 35, 20, 49, 38

Again, you firstly need to order the numbers:

20, 24, 28, 30, 35, 38, 40, 49

And then find the one in the middle:

20, 24, 28, 30, 35, 38, 40, 49

In this example there are actually two numbers that are in the middle. You therefore find the middle of these two numbers by adding them together and then halving the answer:

(30 + 35) ÷ 2
75 ÷ 2 = 37.5

The median for this set of data is 37.5. As there are two numbers that are in the middle, your answer does not appear in the original set of data.

The example below is a little more complicated.

Case study _unit5.6.3 Example: Finding the median 3

Tracy did a survey of the number of cups of coffee her colleagues had drunk during one day. The frequency table shows her results.

Table _unit5.6.15 Table 22
Number of cups of coffee	Frequency
2	1
3	5
4	3
5	4
6	6

First you need to calculate the number of colleagues by adding together the numbers in the frequency column:

1 + 5 + 3 + 4 + 6 = 19

You then need to work out the median or middle value in the number of cups of coffee. To do this you can list the number of cups of coffee in a line:

2 3 3 3 3 3 4 4 4 5 5 5 5 6 6 6 6 6 6

You know there are 19 colleagues, which is an odd number, so you will be able to find the exact midpoint:

9 + 9 = 18

so count in from either side until you reach the 10th number which can be seen as 5 in the list above.

Another way to do this is to calculate the number of colleagues, which you know is 19. You then find the midpoint by adding the numbers in the frequency table:

1 + 5 + 3 = 9

and the exact midpoint is the 10th colleague which the table shows as 4 in the frequency column.

If you look under the Number of cups of coffee column, you can see that the answer is 5 cups of coffee, so the median is 5.

Activity _unit5.6.4 Activity 14: Calculating the median

Now calculate the median of the following:

The ages of a group of students on a course are:
- 16, 44, 32, 67, 25, 18, 22
The heights of a group of children in a gymnastics class are:
- 1.24 m, 1.27 m, 1.20 m, 1.15 m, 1.26 m, 1.17 m
The frequency table below shows the number of televisions that a group of students on a media course have at home.

Table _unit5.6.16 Table 23
Number of televisions	Number of students
0	1
1	4
2	8
3	9
4	3

Calculate the median number of televisions.

Answer

First you need to list the data in order of size so:
- 16, 18, 22, 25, 32, 44, 67
Now find the middle value. In this case there are 7 data values so you will be able to find the exact middle.
- 16, 18, 22, 25, 32, 44, 67
The middle value is 25 so this is the median age of the group of students.
First you need to list the heights in order of size so:
- 1.15 m, 1.17 m, 1.20 m, 1.24 m, 1.26 m, 1.27 m
Now find the middle value. In this case there are 6 data values so first find the middle two values.
- 1.15 m, 1.17 m, 1.20 m, 1.24 m, 1.26 m, 1.27 m
Now add together the two middle values:
- 1.20 m + 1.24 m = 2.44 m
  Then halve the answer:
  2.44 m ÷ 2 = 1.22 m
So the median height of the students is 1.22 m.
First you need to calculate the number of students.
To do this you add up each number in the frequency table:
- 1 + 4 + 8 + 9 + 3 = 25 students
You then need to work out the median or middle value in the number of televisions. To do this you can list the number of televisions in a line:
- 0 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 4 4 4
As you know there are 25 students, you need to find the midpoint which will be 13. Count in until you find the 13th number which is 2 in your list of televisions, so the median is 2.
Another way to do this is to calculate the number of students, which you know is 25. You then find the midpoint as the 13th student using the frequency table. If you count up the ‘Number of students’ column in the frequency table, the 13th value is 2 televisions, so the median is 2.

If you want to see some more examples, or try some for yourself, use the link below:

https://www.mathsisfun.com/ median.html

6.6 Calculating the mode

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

The final type of average to look at is the mode, which is the most common value in a data set. There can sometimes be more than one mode, which occurs when two or more values are equally common. Sometimes there will be no mode as each data value occurs only once. When you calculate the mode you will always find that it is one of the values in your original data set. The mode is also called the modal value.

Case study _unit5.6.4 Example: Finding the mode 1

What is the mode of the following numbers?

3, 7, 5, 6, 4, 5, 6, 5, 7, 5

First group the same numbers together by listing them in size order:

3, 4, 5, 5, 5, 5, 6, 6, 7, 7

It is then easy to identify the modal value as 5 as this number occurs the most.

Box _unit5.6.1

Note: To remember how to calculate this average use Mode = Most.

Case study _unit5.6.5 Example: Finding the mode 2

What is the mode of the following amounts of money?

£8.99 £16.45 £17.50 £36.20 £6.75 £9.35 £12.99 £8.95

First list them in size order from lowest amount to highest:

£6.75 £8.95 £8.99 £9.35 £12.99 £16.45 £17.50 £36.20

You can now see that there is no mode (or modal value) as each amount of money occurs only once.

Case study _unit5.6.6 Example: Finding the mode 3

Below is a set of data showing the number of people attending a yoga class each week over a year. What is the mode?

17 17 13 16 18 15 12 16 16 17 18 13

First list them in size order from lowest amount to highest:

12 13 13 15 16 16 16 17 17 17 18 18

You can now see that two numbers occur the same number of times. 16 and 17 both occur three times, so in this set of data there are two modes or modal values – 16 and 17.

Case study _unit5.6.7 Example: Finding the mode 4

To find the mode from a frequency table you need to find the value with the highest frequency. The results of a survey on a block of flats are shown in the frequency table below. The highest frequency, which can be seen in the ‘Number of flats’ column, is 18. This means that the mode or modal number of occupants is 3.

Table _unit5.6.17 Table 24
Number of occupants	Number of flats
0	2
1	9
2	13
3	18
4	6

Activity _unit5.6.5 Activity 15: Calculating the mode

Now calculate the mode of the following:

3, 6, 5, 7, 3, 5, 6, 6, 3, 4, 9, 6
13, 19, 11, 28, 17, 29, 16, 24, 15, 18
81 cm, 53 cm, 74 cm, 62 cm, 53 cm, 70 cm, 81 cm, 74 cm, 42 cm, 90 cm
The table below shows the number of tries scored by a school rugby team during one month. What is the modal number of tries scored?

Table _unit5.6.18 Table 25(a)
Number of tries	Frequency
0	5
1	8
2	6
3	3

Answer

First list them in size order from lowest amount to highest:
- 3, 3, 3, 4, 5, 5, 6, 6, 6, 6, 7, 9
It is then easy to identify the modal value as 6 as this number occurs the most.
First list them in size order from lowest amount to highest:
- 11, 13, 15, 16, 17, 18, 19, 24, 28, 29
You can now see that each value occurs only once so there is no mode/modal value.
First list them in size order from lowest amount to highest:
- 42 cm, 53 cm, 53 cm, 62 cm, 70 cm, 74 cm, 74 cm, 81 cm, 81 cm, 90 cm
You can now see that three numbers occur the same number of times. 53 cm, 74 cm and 81 cm occur twice, so in this set of data there are three modes or modal values.
To find the mode you need to look for the highest frequency in the table. In this case it is 8 which shows that the modal number of tries scored is 1.

Table _unit5.6.19 Table 25(b)
Number of tries	Frequency
0	5
1	8
2	6
3	3

6.7 Choosing the best average

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

For some sets of data it may be better to use one type of average over another as it will be more representative of the data type.

Here are some of the advantages and disadvantages of each type of average.

Box _unit5.6.2 Mean

Advantages

Uses all the data values.

Disadvantages

May not always be one of the values in the set of data or a value that does not make sense for the data, e.g. 1.6 people.
The mean may not be useful for a set of data which has a value a lot higher or lower than the others, e.g. If you were to include the salary of the Managing Director with the wages of shop floor staff when calculating an average salary, it is likely that the mean would be distorted by the higher salary of the Managing Director, which means that it would not represent the data very well.

Box _unit5.6.3 Median

Advantages

It is the middle value so is not affected by very high or very low values. It is a useful type of average for data sets with such values.

Disadvantages

Sometimes it will not be one of the values in the data set.

Box _unit5.6.4 Mode

Advantages

It will always be one of the values in the data set (if there is a mode).
It is very good for certain types of data, e.g. finding the most common shoe size.

Disadvantages

There may not be a mode.
There may be several modes.
It may be at one end of the data distribution.

Now have a go at finding the mean, median and mode.

Activity _unit5.6.6 Activity 16: Finding different averages

A bridal shop records the sizes of wedding dresses that it sells in one month. The table below shows the results.

Find the following:
- a.the mean
- b.the median
- c.the mode.
Which of the averages gives the most useful information for this data set?

Table _unit5.6.20 Table 26(a)
Dress size	No. of dresses sold
8	2
10	8
12	11
14	12
16	5
18	2

Answer

(a) The mean
- First, work out the total number of dresses sold by multiplying the dress size column by the number of dresses sold.
- Add a frequency column to display your calculations.
- Next, work out the totals for the number of dresses sold column and the frequency column.

Table _unit5.6.21 Table 26(b)
Dress size	No. of dresses sold	Frequency
8	2	8 × 2 = 16
10	8	10 × 8 = 80
12	11	12 × 11 = 132
14	12	14 × 12 = 168
16	5	16 × 5 = 80
18	2	18 × 2 = 36
Totals	40	512

- Finally calculate the mean by dividing the frequency by the number of dresses sold:
- 512 ÷ 40 = 12.8
- Mean = 12.8

(b) The median
- First you need to calculate the total number of dresses sold. To do this add up each number in the frequency table:
  2 + 8 + 10 + 13 + 5 + 2 = 40 dresses sold.
- Now find the middle value by listing the quantity of each size of dress from smallest to biggest:

- 8 8 10 10 10 10 10 10 10 10 12 12 12 12 12 12 12 12 12 12 12 14 14 14 14 14 14 14 14 14 14 14 14 16 16 16 16 16 18 18
- The midpoint of the 40 dresses sold is between values 20 and 21 so you need to find both of these values. In this example they are 12 and 12.
  You calculate the median by adding 12 + 12 and dividing by 2:
  12 + 12 = 24
  24 ÷ 2 = 12
  As you can see, in this example the median is size 12.
- A quicker way to do this is to calculate the number of dresses, which you know is 40.
- You then use the frequency table to find the midpoint which is between the 20th and the 21st dress size. If you count up the number of dresses sold in the frequency table:
  2 + 8 + 11 = 21
  you can see that the 20th and 21st values fall within size 12, so the median is size 12.
  Median = 12

(c) The mode
- To find the mode you need to look for the highest number of dresses sold. In this case it is 12 which shows that the mode dress size is 14.
  Mode = 14

Table _unit5.6.22 Table 26(c)
Dress size	No. of dresses sold
8	2
10	8
12	11
14	12
16	5
18	2

In this case the mean is size 12.8 which does not exist as a dress size so this is not useful.
The median result is size 12 which is a dress size, but it is still not the most commonly sold dress size.
The mode is dress size 14 which is the most commonly sold dress size and so this gives the most useful information.

Well done! You have now learned all you need to know about mean, median, mode and range. The final part of this section, before the end-of-course quiz, looks at probability.

Summary

In this section you have learned:

that there are different types of averages that can be used when working with a set of data – mean, median and mode
range is the difference between the largest data value and the smallest data value and is useful for comparing how consistently someone or something performs
mean is what is commonly referred to when talking about the average of a data set
how to find the mean from both a single data set and also a set of grouped data
what the median of a data set is and how to find it for a given set of data
what the mode of a data set is and how to find it for a given set of data
how to choose the ‘best’ type of average for a given set of data.

7 Probability

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

You will use probability regularly in your day-to-day life:

Should you take an umbrella out with you today?
What are the chances of the bus being on time?
How likely are you to meet your deadline?

Figure _unit5.7.1 Figure 26 Probability – the odds are you are already using it

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A probability cartoon of two pupils outside the Principal’s office. The caption reads ‘I wish we hadn’t learned probability ‘cause I don’t think our odds are good’.

Figure 26 Probability – the odds are you are already using it

Probability is all about how likely, or unlikely, something is to happen. When you flip a coin for example, the chances of it landing on heads is or 50% or 0.5 (do you remember from your work in Session 1 how fractions, decimals and percentages can be converted into one another?).

The probability, or likelihood, of an event happening is perhaps most easily expressed as a fraction to begin with. Then, if you want to express it as a percentage or a decimal you can just convert it.

Let’s look at an example.

Case study _unit5.7.1 Example: Chocolate probability

A box of chocolates contains 15 milk chocolates, 5 dark chocolates and 10 white chocolates. If the box is full and you choose a chocolate at random, what is the likelihood of choosing a dark chocolate?

Method

There are 5 dark chocolates in the box. There are 15 + 5 + 10 = 30 chocolates in the box altogether.

The probability of choosing a dark chocolate is therefore:

Extract _unit5.7.1

You could also be asked the probability of choosing either a dark or a white chocolate. For this you just need the total of dark and white chocolates:

Extract _unit5.7.2

5 + 10 = 15

The total number of chocolates in the box remains the same so the likelihood of choosing a dark or a white chocolate is:

Extract _unit5.7.3

You could even be asked what the likelihood is of an event not happening. For example, the likelihood that you will not choose a white chocolate. In this case, the total number of chocolates that are not white is 15 + 5 = 20.

Again, the total number of chocolates in the box remains the same and so the probability of not choosing a white chocolate is:

Extract _unit5.7.4

Now have a go yourself by completing the short activity below.

Activity _unit5.7.1 Activity 17: Calculating probability

You buy a packet of multi-coloured balloons for a children’s party. You find that there are 26 red balloons, 34 green balloons, 32 yellow balloons and 28 blue balloons.
You take a balloon out of the packet without looking. What is the likelihood of choosing a green balloon?
Give your answer as a fraction in its simplest form.
At the village fete, 350 raffle tickets are sold. There are 20 winning tickets. What is the probability that you will not win the raffle?
Give your answer as a percentage rounded to two decimal places.

Answer

There are 34 green balloons. The total number of balloons is 26 + 34 + 32 + 28 = 120.
The likelihood of choosing a green balloon is therefore:
- = in its simplest form.
If there are 20 tickets that are winning ones, there are 350 − 20 = 330 tickets that will not win a prize.
As a fraction this is:
To convert to a percentage, you do 330 ÷ 350 × 100 = 94.29% rounded to two d.p.

You sometimes need to calculate the probability of more than one event occurring. In this case, the events might be:

independent – which means that the outcome of one event does not affect the other
dependent – which means that the outcome of one event does affect the other.

In either case, you can use tree diagrams or tables to help you to picture and solve these problems.

Case study _unit5.7.2 Example: Gender probability

If a couple has two children, the gender of the first child will not affect the gender of the second child. All possibilities can be shown in the form of a table:

B = boy
G = girl

Table _unit5.7.1 Table 27 First and second child gender probability table
		First child
		B	G
Second child	B	BB	GB
Second child	G	BG	GG

Alternatively, it can be shown as a tree diagram:

Figure _unit5.7.2 Figure 27 A tree diagram showing first and second child gender outcomes

Show description|Hide description

The graphic shows three columns across the top labelled (from left to right) ‘1st child’, ‘2nd child’ and ‘Outcome’. Under the ‘1st child’ column are the letters B (boy) and G (girl). Two lines go from ‘B’ under the ‘1st child’ column to ‘B’ and ‘G’ under the ‘2nd child’ column. Two lines go from ‘G’ in the ‘1st child’ column to another ‘B’ and ‘G’ in the ‘2nd child’ column. the ‘Outcome’ column displays the possible outcomes, each under the other: BB, BG, GB, GG.

Figure 27 A tree diagram showing first and second child gender outcomes

Both the table and the tree diagram show that there are four possibilities:

BB – boy then another boy
BG – boy then a girl
GB – girl then a boy
GG – girl then another girl

Out of the possibilities there is 1 in 4 or (1 quarter/25%) chance of having 2 boys or 2 girls.

There is a 2 in 4 or (/1 half/50%) chance of having one child of each gender.

Activity _unit5.7.2 Activity 18: Using diagrams and tables to calculate probability

Complete the missing details in the following tree diagram:
Figure _unit5.7.3 Figure 28 A tree diagram showing first and second coin toss outcomes
Show description|Hide description
The graphic shows three columns across the top labelled (from left to right) ‘1st coin’, ‘2nd coin’ and ‘Outcome’. Under the ‘1st coin’ column are the letters H (heads) and T (tails). Two lines go from ‘H’ under the ‘1st coin’ column to ‘H’ and ‘T’ under the ‘2nd coin’ column. Another two lines go from ‘T’ in the ‘1st coin’ column to another ‘H’ and ‘T’ in the ‘2nd coin’ column. The ‘Outcome’ column displays ‘?(a)’, under is ‘HT’, under this is ‘?(b)’ and under this is ‘?(c)’.
Figure 28 A tree diagram showing first and second coin toss outcomes
Draw a table showing all of the possibilities when 2 coins are tossed.
What is the probability of getting 2 tails?

Answer

- a.HH
- b.TH
- c.TT
Your table should look like the one below.

Table _unit5.7.2 Table 28 Coin toss probability table
		First coin
		H	T
Second coin	H	HH	TH
Second coin	T	HT	TT

Out of the possibilities there is 1 in 4 or (1 quarter/25%) chance of getting 2 tails.

Remember that when you are checking your answers, you may have gone about the question in a different way. In a real exam it is always important to show your working as even if you don’t arrive at the correct answer you can still gain marks.

You’ve now completed Session 4 of your course, congratulations!

Summary

In this section you have learned:

that the probability of an event is how likely or unlikely that event is to happen and that this can be expressed as a fraction, decimal or percentage.
how to use a table or tree diagram to show the different outcomes of two or more events.

8 End-of-course quiz

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Now it’s time to complete end-of-course quiz (Session 4 compulsory badge quiz). It’s similar to previous quizzes, but in this one there will be 15 questions.

End-of-course quiz

Open the quiz in a new window or tab then come back here when you’re done.

Remember, this quiz counts towards your badge. If you’re not successful the first time, you can attempt the quiz again in 24 hours.

9 Session 4 summary

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Well done! You have now completed ‘Handling data’, the fourth and final session of the course. If you have identified any areas that you need to work on, please ensure you refer back to this section of the course.

You should now be able to:

identify different types of data
create and use tally charts, frequency tables and data collection sheets to record information
draw and interpret bar charts, pie charts and line graphs
understand there are different types of averages and be able to calculate each type
understand that probability is about how likely an event is to happen and the different ways that it can be expressed.

All of the skills listed above will help you when booking a holiday, budgeting, reading the news or analysing data at work.

You are now ready to test the knowledge and skills you’ve learned throughout each section in the end-of-course quiz (Session 4 compulsory badge quiz). Good luck!

10 Bringing it all together

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Congratulations on completing Everyday maths 2. We hope you have enjoyed the experience and now feel inspired to develop your maths skills further.

Throughout this course you have developed your skills within the following areas:

understanding and using whole numbers and decimal numbers, and understanding negative numbers in the context of money and temperature
solving problems requiring the use of the four operations and rounding answers to a given degree of accuracy
understanding and using equivalences between common fractions, decimals and percentages
working out simple and more complex fractions and percentages of amounts
calculating percentage change
adding, subtracting, multiplying and dividing decimals up to three decimal places
solving ratio problems where the information is presented in a variety of ways
understanding the order of operations and using this to work with formulas
solving problems requiring calculations with common measures, including money, time, length, weight, capacity and temperature
converting units of measure in the same system and those in different systems
extracting and interpreting information from tables, diagrams, charts and graphs
collecting and recording discrete data, and organising and representing information in different ways
finding the mean, median, mode and range for sets of data
using data to assess the likelihood of an outcome and expressing this in different forms
working with area, perimeter and volume, scale drawings and plans.

11 Next steps

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

If you would like to achieve a more formal qualification, please visit one of the centres listed below with your OpenLearn badge. They’ll help you to find the best way to achieve the Level 2 Essential Skills Wales qualification in Application of Number, which will enhance your CV.

Coleg Cambria • https://www.cambria.ac.uk/ • 0300 30 30 007
Addysg Oedolion Cymru | Adult Learning Wales • https://www.adultlearning.wales/ • 03300 580845
Coleg Gwent • https://www.coleggwent.ac.uk/ • 01495 333777
NPTC Group of Colleges • https://www.nptcgroup.ac.uk/

Introduction

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

If you want more practice with your maths skills, please contact your local college to discuss the options available.

By the end of this session you will be able to:

use the four operations to solve problems in context
understand rounding and look at different ways of doing this
write large numbers in full and shortened forms
carry out calculations with large numbers
carry out multistage calculations
solve problems involving negative numbers
define some key mathematical terms (multiple, lowest common multiple, factor, common factor and prime number)
identify lowest common multiples and factors
use fractions, decimals and percentages and convert between them
solve different types of ratio problems
make substitutions within given formulas to solve problems
use inverse operations and estimations to check your calculations.

Download this video clip.Video player: bltl_a69_numbers.mp4

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1 Four operations

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Figure _unit5.13.1 Figure 1 Fruit maths puzzle

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Figure 1 Fruit maths puzzle

Addition (+)
- This operation is used when you want to find the total, or sum, of two or more amounts.
Subtraction (−)
- This operation is used when you want to find the difference between two amounts or how much of something you have left after a quantity is used – for example, if you want to find the change owed after spending an amount of money.
Multiplication (×)
- This operation is used to calculate multiple amounts of the same number. For example, if you wanted to know the cost of 16 plants costing £4.75 each you would multiply £4.75 by 16:
  £4.75 × 16 = £76
Division (÷)
- Division is used when sharing or grouping items. For example, if a group of 6 friends won £765 on the lottery and you wanted to know how much each person would get you would divide £765 by 6:
  £765 ÷ 6 = £127.50

Activity _unit5.13.1 Activity 1: Operation choice

Each of the four questions below uses one of the four operations. Match the operation to the question.

Using the following two lists, match each numbered item with the correct letter.

a.Fourteen members of the same family go on holiday together. They each pay £155. What is the total cost of the holiday?
b.You go to the local café and buy a coffee for £2.35, a tea for £1.40 and a croissant for £1.85. How much do you spend?
c.You need to save £306 for a holiday. You have 18 months to save up that much money. How much do you need to save per month?
d.You make an insurance claim worth £18 950. The insurance company pays you £12 648. What is the difference between what you claimed and what you actually received?

The correct answers are:

1 = c
2 = a
3 = d
4 = b

Try these questions to make sure that you are confident using each of the four operations.

Activity _unit5.13.2 Activity 2: Calculations with whole numbers

1245 + 654
187 + 65 401
1060 − 264
2000 − 173
543 × 19
1732 × 46
1312 ÷ 8
1044 ÷ 12
Stuart is saving £225 per month for a new car. How much would he have saved in:
- a.1 year
- b.2 years
A concert venue has 2190 seats arranged into 15 sections. How many seats are in each section?
A college sells 325 tickets to the charity ball at £12 each. How much money do they make from ticket sales?

Answer

1245 + 654 = 1899
187 + 65 401 = 65 588
1060 − 264 = 796
2000 − 173 = 1827
543 × 19 = 10 317
1732 × 46 = 79 672
1312 ÷ 8 = 164
1044 ÷ 12 = 87
- a.Stuart is saving £225 per month for 1 year or 12 months so the calculation is 225 × 12 = £2700.
- b.Stuart saves £2700 in 1 year so to work out the amount saved in 2 years the calculation is 2 × £2700 = £5400. Alternatively, you could add £2700 + £2700 = £5400.
To calculate this you divide the number of seats by the number of sections.
So, 2190 ÷ 15 = 146. So there are 146 seats in each section.
To calculate this you multiply the number of tickets by the cost per ticket.
So, 325 × 12 = 3900. So the college will make £3900 from ticket sales.

1.1 Expressing a remainder as a decimal

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

To split a prize of £125 between 5 friends you would do this calculation:

£125 ÷ 5 and get the answer £25.

This is a convenient, exact amount of money. However, often when you perform calculations, especially those involving division, you do not always get an answer that is suitable for the question.

Figure _unit5.13.2 Figure 2 Expressed as a decimal: 125 ÷ 4

We can use the same principal with any whole number, adding as many zeros after the decimal point as required. Look at the following example.

A teacher wants to share 35 kg of clay between 8 groups of students. How much clay will each group get?

Figure _unit5.13.3 Figure 3 Expressed as a decimal: 35 ÷ 8

You can see that each group would get 4.375 kg of clay.

Activity _unit5.13.3 Activity 3: Expressing a remainder as a decimal

Work out the answers to the following without using a calculator.

178 ÷ 4
212 ÷ 5
63 ÷ 8
227 ÷ 4

Answer

44.5
42.4
7.875
56.75

1.2 Interpreting answers when dividing

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

After carrying out a division calculation you may not have an answer that is suitable.

Box _unit5.13.1

Activity _unit5.13.4 Activity 4: Interpreting answers

Calculate the answers to the following. Decide whether the answers need to be adjusted up or down after calculation of the division sum.

Apples are being packed into boxes of 52. There are 1500 apples that need packing. How many boxes are required?
A bag of flour contains 1000 g. Each batch of cakes requires 150 g of flour. How many batches can you make?
A child gets £2.50 pocket money each week. They want to buy a computer game that costs £39.99. How many weeks will they need to save up in order to buy the game?
A length of copper pipe measures 180 cm. How many smaller pieces that each measure 40 cm can be cut from the pipe?

Answer

1500 ÷ 52 = 28.846 which must be adjusted up to 29 boxes.
1000 ÷ 150 = 6.666 which must be adjusted down to 6 batches.
£39.99 ÷ £2.50 = 15.996 which must be adjusted up to 16 weeks.
180 ÷ 40 = 4.5 which must be adjusted down to 4 pieces.

1.3 Dealing with decimals

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

We often deal with decimals in everyday life, for example, calculations involving money.

Activity _unit5.13.5 Activity 5: Calculations with decimals

54.865 + 4.965 + 23.519
6.938 − 5.517
25 + 0.258
54 − 0.65
5.632 × 2.4
1.542 × 1.9
42.4 ÷ 4
39.45 ÷ 1.5
0.48 ÷ 0.025
A factory orders 425 gaskets that weigh 2.3 g each. What is the total weight of the gaskets?
A dispenser holds 15.5 litres of water. How many full 0.2 litre cups of water can you get from one dispenser?
Ahmed and Lea take part in the long jump. Their results are as follows:
- Ahmed 5.501 m
  Lea 5.398 m
  Who jumped furthest and by how much?

Answer

83.349
1.421
25.258
53.35
13.5168
2.9298
10.6
26.3
19.2
977.5 g
77.5 (77 full cups)
Ahmed jumped the furthest by 0.103 m

Summary

In this section you have:

recapped how to carry out calculations with whole numbers and decimals
learned to adjust answers where needed
learned how to express a remainder as a decimal.

2 Dealing with large numbers

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

It is important to be able to carry out calculations with numbers of any size. Large numbers can be written in different ways e.g.

Extract _unit5.14.1

1 200 000 (one million, two hundred thousand) or it can be written as 1.2 million.

Here is another example:

Extract _unit5.14.2

4 250 000 000 (four billion, two hundred and fifty million) is 4.25 billion.

It is often easier to deal with very large numbers when they are written as decimals.

Notice how the decimal is placed after the whole millions or billions.

Hint: A billion is a thousand million.

Using a place value grid can help you to read large numbers as it groups the digits for you, making the whole number easier to read.

Notice how the numbers above are written in this place value grid.

Table _unit5.14.1 Table 1
Billion	Million			Thousand
Billions	Hundreds of millions	Tens of millions	Millions	Hundreds of thousands	Tens of thousands	Thousands	Hundreds	Tens	Units
			1	2	0	0	0	0	0
4	2	5	0	0	0	0	0	0	0

Activity _unit5.14.1 Activity 6: Rounding large numbers

The following table gives the population of countries.

Round each population to the nearest million and write the figure in shortened form, using decimals where needed.

Table _unit5.14.2 Table 2(a)
Country	Population
UK	66 959 016
China	1 420 062 022

Answer

Table _unit5.14.3 Table 2(b)
Country	Population	Population rounded	Shortened form
UK	66 959 016	67 000 000	67 million
China	1 420 062 022	1 420 000 000	1.42 billion

2.1 Calculations with large numbers

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

The best way to get used to these types of calculations is to go straight into an example.

Case study _unit5.14.1 Example: Calculations with large numbers

Calculate the total population of Malta (0.4 million) and Cyprus (1.2 million).

Method 1

Work in shortened form:

1.2 + 0.4 = 1.6 million

Method 2

Write the numbers in full:

1 200 000 + 400 000 = 1 600 000 (1.6 million)

Activity _unit5.14.2 Activity 7: Calculations with large numbers

Calculate the total turnover for Cambria Trading over the first quarter (3 months).

Table _unit5.14.4 Table 3 Cambria Trading turnover
Month	Profit (£ million)
January	1.2
February	0.9
March	0.85
April	1.1
May	1.02
June	0.87
July	1.19
August	0.98
September	1.05
October	1.08
November	1.8
December	1.65

Calculate the turnover of the last quarter.
Calculate the difference in turnover between the first and last quarters.
- a.Which month had the largest turnover?
- b.Which month had the smallest turnover?
- c.What is the difference between the largest and smallest turnovers?

Answer

Profit in 1st quarter 1.2 + 0.9 + 0.85 = £2.95 million
Profit in last quarter 1.08 + 1.8 + 1.65 = £4.53 million
The difference is 4.53 − 2.95 = £1.58 million
- a.November had the largest turnover at £1.8 million.
- b.March had the smallest at £0.85 million.
- c.The difference is 1.8 − 0.85 = £0.95 million.

Summary

In this section you have learned how to:

write large numbers in full and shortened forms
round large numbers
add and subtract large numbers.

3 Rounding

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Figure _unit5.15.1 Figure 4 Rounding up and down

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Illustration of an up arrow and a down arrow

Figure 4 Rounding up and down

You will now explore each of these examples in more detail and practise your rounding skills in context.

3.1 Rounding to a degree of accuracy

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Watch the short video below to see an example of how to round to 1, 2 and 3 decimal places.

Download this video clip.Video player: bltl_1_2_3_rounding.mp4

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Transcript

This is the second decimal place. Look at the number next to it. Is it greater than or equal to 5? 6 is more than 5, so you need to add 1. The amount to pay is £29, 32 pence.

Can you round to three decimal places? Have a go with this number: 412.5762. The number next to the third decimal place is 2, which is less than 5. This means the correctly rounded number is 412.576.

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Remember this rounding rhyme to help you:

Figure _unit5.15.2 Figure 5 A rounding rhyme

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A rounding rhyme. The text reads: Underline the digit, look next door. If it’s 5 or greater, add one more. If it’s less than 5, leave it for sure. Everything after is a zero, not more.

Figure 5 A rounding rhyme

Activity _unit5.15.1 Activity 8: Rounding skills

Practise your rounding skills by completing the below.

What is 24.638 rounded to one decimal place?
What is 13.4752 rounded to two decimal places?
What is 203.5832 rounded to two decimal places?
What is 345.6794 rounded to three decimal places?
What is 3.65 rounded to the nearest whole number?
What is £199.755 to the nearest penny?
What is £37.865 to the nearest pound?
What is 61.607 kg to the nearest kg?

Answer

24.6
13.48
203.58
345.679
4
£199.76
£38
62 kg

3.2 Rounding to approximate an answer

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Activity _unit5.15.2 Activity 9: Approximation

Calculate the following using a calculator and use estimation to check your answers.

On 5th March 2019, 190 people matched 5 numbers and won £1650 each. What was the total prize fund?
Swansea AFC’s home ground is the Liberty Stadium which holds 21 088 fans. Cardiff City FC plays at the Cardiff City Stadium which holds 33 316 fans.
What is the difference in capacity at these grounds?

Answer

The actual answer is £313 500 (190 × 1650)
Estimate 200 × 1700.
2 × 17 = 34 so 200 × 1700 = 340 000 so the answer is sensible.
The actual answer is 12 228 (33 316 − 21 088)
Estimate 33 000 − 21 000 = 12 000 so the answer is sensible.

Summary

In this section you have learned:

how and when to use rounding to approximate an answer to a calculation
how to round an answer to a given degree of accuracy – e.g. rounding to two decimal places.

4 Multistage calculations

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Often in daily life you will come across problems that require more than one calculation to reach the final answer.

Case study _unit5.16.1 Example: Multistage calculations

Four friends are planning a holiday. The table below shows the costs:

Table _unit5.16.1 Table 4
Item	Price
Flight (return)	£305 per person
Taxes	£60 per person
Hotel	£500 per room. 2 people per room
Taxi to airport	£45

The friends will be sharing the total cost equally between them. How much do they each pay?

Method

First we use multiplication to find the cost of items that we need more than one of:

Flights = £305 × 4 = £1220
Taxes = £60 × 4 = £240
Hotel = 2 rooms required for 4 people = £500 × 2 = £1000

Now we use addition to add these totals together along with the taxi fare:

£1220 + £240 + £1000 + £45 = £2505

Finally, we need to use division to find out how much each person pays:

£2505 ÷ 4 =£626.25 each

Activity _unit5.16.1 Activity 10: Multistage calculations

Your current mobile phone contract costs you £24.50 per month.
You are considering changing to a new provider. This provider charges £19.80 per month along with an additional, one off connection fee of £30.
How much will you save over the year by switching to the new provider?
You are going on holiday and you have decided to stay in a cottage in North Wales for 7 nights.
There will be 12 of you staying and the total cost of the cottage you have chosen is £460 per night. If you split the cost equally, how much will each of you pay?

Answer

First calculate the cost with the current provider
monthly cost × 12
£24.50 × 12 = £294 (current provider)
Second, calculate the cost of the new provider.
To do this you need to calculate the total monthly costs:
- £19.80 × 12 = £237.60
and then add on the one off connection fee of £30:
- £237.60 + £30 = £267.60 (new provider)
Finally you can calculate the difference between the two providers:
- £294 − £267.60 = £26.40 saved
To do your calculation, you need to work out the cost for 7 nights. Once you have done this, you can divide the total by 12:
- 460 × 7 = £3220
  £3220 ÷ 12 = £268.33 (rounded to two d.p.)
  Here we would probably round this amount up to £268.34 per person to make sure the full cost is covered.

Summary

In this section you have:

applied the four operations to solve multistage calculations.

5 Negative numbers

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Negative numbers come into play in two main areas of life: money and temperature. Watch the animations below for some examples.

Download this video clip.Video player: bltl_1_3_negative.mp4

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Transcript

Similarly, you'll be used to seeing winter weather forecasts predicting temperatures of -10 degrees Celsius. This means the temperature will be 10 degrees below zero degrees Celsius.

For example, you'd like to know the answer to the sum [6 - 10]. Starting at number 6 on a thermometer and travelling down, it's easy to see that subtracting 10 will result in a number less than zero.

What is the answer to the sum [-5 + 12]? This time, we're starting below zero and travelling up the thermometer by 12 places, giving an answer of 7.

Can you work out the answer to the sum [-1 - 8]? In this example, we start below zero and the answer is also below zero: -9. Now try the next activity for more practise with negative numbers.

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Activity _unit5.17.1 Activity 11: Negative and positive temperature

The table below shows the temperatures of cities around the world on a given day

Table _unit5.17.1 Table 5
London	Oslo	New York	Kraków	Delhi
4˚C	−12C	7˚C	−3˚C	19˚C

- a.Which city was the warmest?
- b.Which city was the coldest?
- c.What is the difference in temperature between the warmest and coldest cities?

Answer

- a.Delhi was the warmest city as it has the highest positive temperature.
- b.Oslo was the coldest city as it has the largest negative temperature.
- c.The difference between the temperatures in these cities is 31˚C.
  From 19˚C down to 0˚C is 19˚C and then you need to go down a further 12˚C to get to −12˚C.

Look at this bank statement.
Figure _unit5.17.1 Figure 6 A bank statement
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A bank statement with four columns: Date, Description, Amount and Balance. There are six lines under the column headings showing money paid in and out. (Date column): 09 Oct, 11 Oct, 15 Oct, 20 Oct, 21 Oct, 25 Oct. (Description column): Bank Transfer, Direct Debit, Automated Pay In, Bank Bank Transfer, Direct Debit, Bank Transfer. The (Amount column) is blank. (Balance column): 100.00, −20.00, 70.00, 100.00, −50.00, 200.00.
Figure 6 A bank statement
- a.On which days was Sonia Cedar overdrawn, and by how much?
- b.How much money was withdrawn between 9 and 11 of October?
- c.How much was added to the account on 15 October?

Answer

- a.The minus sign (−) indicates that the customer is overdrawn, i.e. owes money to the bank.
  The amount shows how much they owe. So Sonia Cedar was overdrawn on 11 October by £20 and by £50 on 21 October.
- b.£120 was withdrawn on 11 October.
  The customer had £100 in the account and must have withdrawn another £20 (i.e. £100 + £20 = £120 in total) in order to be £20 overdrawn.
- c.The customer owed £20 and is now £70 in credit, so £90 must have been added to the account.

Look at the table below showing a company’s profits over 6 months.
Hint: a negative profit means that the company made a loss.

Table _unit5.17.2 Table 6
Month	Profit (£000)
January	166
February	182
March	−80
April	124
May	98
June	−46
Balance

- a.Which month had the greatest profit?
- b.Which month has the greatest loss?
- c.What was the overall balance for the six months?
  Hint: start by calculating the total profits and the total losses.

Answer

- a.February had the largest profit with £182 000 (remember to look at the column heading which shows that the figures are in 000s – thousands).
- b.March showed the greatest loss at £80 000.
- c.To calculate the overall balance you need to first calculate the total profits and the total losses. To calculate the profits you need to do this calculation:
- 166 + 182 + 124 + 98 = 570
So the profit was £570 000.
Next you need to calculate the total losses; two months showed a loss so you need to add these values:
- 80 + 46 = 126 so the losses over the six months were £126 000.
Now you can calculate the overall balance by subtracting the losses from the profits:
- £570 000 − £126 000 = £444 000
This is a positive value so it means the company made an overall profit of £444 000.

Summary

In this section you have:

learned the two main contexts in which negative numbers arise in everyday life – money (or debt!) and temperature
practised working with negative numbers in these contexts.

6 Mathematical terms

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

It is important to know the meaning of the following terms:

multiples
lowest common multiple
factors
common factors
prime numbers

Multiples

A multiple of a number can be found by multiplying that number by any whole number e.g. multiples of 2 include 2, 4, 6, 8, 10 etc. (all are in the 2 times table).

Box _unit5.18.1

Note: To check if a number is a multiple of another, see if it divides exactly into the multiple, e.g. to see if 3 is a multiple of 81 do 81 ÷ 3 = 27. It divides exactly so 81 isa multiple of 3.

Lowest common multiple

In maths, we sometimes need to find the lowest common multiple of numbers.

The lowest common multiple (LCM) is simply the smallest multiple that is common to more than one number.

Case study _unit5.18.1 Example: Lowest common multiple of 3 and 5

Hint: when looking for multiples, it is easiest to start by listing the multiples of the highest number first. This saves you going any further than you need to with the list.

The first few multiples of 5 are:

5, 10, 15, 20, 25, 30 etc.

The first few multiples of 3 are:

3, 6, 9, 12, 15, 18, 21 etc.

You can see that the lowest number that is a common multiple of 3 and 5 is 15.

Activity _unit5.18.1 Activity 12: Finding the lowest common multiple

Find the lowest common multiple of:

6 and 12
2 and 7

Answer

The lowest common multiple of 6 and 12 is 12:
Multiples of 12:
12, 24, 36, 48, 60 etc.
Multiples of 6:
6, 12, 18, 24, 30 etc.
You can see from the list that 24 is also a common multiple of 6 and 12, but 12 is the lowest common multiple.
The lowest common multiple of 2 and 7 is 14:
Multiples of 7:
7, 14, 21, 28, 35, 42 etc.
Multiples of 2:
2, 4, 6, 8, 10, 12, 14 etc.

Factors, common factors and prime numbers

Case study _unit5.18.2 Example: What are the factors of 8?

8 × 1 = 8

2 × 4 = 8

So the factors of 8 are 1, 2, 4 and 8.

Activity _unit5.18.2 Activity 13: Finding factors

What are the factors of 54?
What are the factors of 165?

Answer

The factors of 54 are 1, 2, 3, 6, 9, 18, 27 and 54
The factors of 165 are 1, 3, 5, 11, 15, 33, 55 and 165

A common factor is a factor that goes into more than one number. For example, 4 is a common factor of 8 and 12 because it divides exactly into both numbers.

Prime Numbers

A prime number is a number which only has 2 factors: 1 and itself.

The prime numbers between 1 and 20 are 2, 3, 5, 7, 11, 13, 17 and 19.

Box _unit5.18.2

Note:

1 is not a prime number as it only has one factor.
2 is the only even prime number.

Summary

In this section you have:

learned some key mathematical terms: multiple, lowest common multiple, factor, common factor and prime number
identified the lowest common multiple
identified factors.

7 Fractions

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Figure _unit5.19.1 Figure 7 A poster advertising a sale

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An illustration of a sale poster reading ‘Sale, many lines 1/3 off original price’.

Figure 7 A poster advertising a sale

There are several different elements to working with fractions. First you will look at simplifying fractions.

7.1 Simplifying fractions

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Watch the video below which looks at how to simplify fractions before having a go yourself in Activity 14.

Download this video clip.Video player: bltl_1_4_2_simplifying_fractions.mp4

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Transcript

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Activity _unit5.19.1 Activity 14: Simplifying fractions

Show the following fractions in simplest form, where possible:

Answer

a. =
b. =
c. =
d. =
e. can’t be simplified
f. =

Next you’ll look at expressing a quantity of an amount as a fraction.

7.2 Writing a quantity of an amount as a fraction

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Sometimes you will need to show one amount as a fraction of another. This might sound complicated, but it’s actually very logical. Look at the examples below.

Case study _unit5.19.1 Example 1: Fraction of an amount

In Figure 8, what fraction of Smarties are red?

Figure _unit5.19.2 Figure 8 Smarties in different colours

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A quantity of Smarties in different colours. 4 are red; 6 are blue; 5 are pink; 3 are green; 6 are yellow; 2 are orange; 4 are purple.

Figure 8 Smarties in different colours

Extract _unit5.19.1

You may well be asked to give your answer as a fraction in its simplest form, so always check to see if you can simplify your answer. In this case will simplify to .

Case study _unit5.19.2 Example 2: Fraction of an amount

250 g of flour is taken from a 1 kg bag. What fraction is this?

Hint: there are 1000 g in a kg.

To express quantities as fractions, the top and bottom numbers need to be in the same units, so here you need to make sure that you express both the top and bottom values in grams:

The flour removed is already expressed in grams: 250 g
The total amount is in kilograms so you need to convert to grams: 1 kg = 1000 g

Now write the amount taken over the total amount to express as a fraction:

(250 g of flour out of the 1000 g bag has been taken or used)

Then cancel down (or simplify) if possible:

So of the flour has been used.

Activity _unit5.19.2 Activity 15: Expressing one number as a fraction of another

What fraction of a kilogram is:
- a.100 g
- b.750 g
- c.640 g
- d.20 g

Answer

100 g = 1000 g, so:
- a.100 g = of a kilogram
- b.750 g = of a kilogram
- c.640 g = of a kilogram
- d.20 g = = of a kilogram

What fraction of an hour is:
- a.15 minutes
- b.20 minutes
- c.35 minutes
- d.48 minutes

Answer

1 hour = 60 minutes so:
- a.15 minutes = = of an hour.
- b.20 minutes = = of an hour.
- c.35 minutes = = of an hour.
- d.48 minutes = = of an hour.

A farmer takes 120 eggs to the local farmer’s market. She has 24 eggs left at the end of the day. What fraction of the eggs are left?

Answer

are left. This cancels down to , so of the eggs are left.

A class of students sit a test. 18 pass and 12 fail. What fraction passed the test?

Answer

Work out the total number of students by adding the number who passed to those who failed 18 + 12 = 30. Now work out the fraction that passed:
- (18 out of 30 students passed)
Now cancel down:
- = .
  So passed the test.

Mary bought her car for £12 500. When she goes to trade it in she is offered £8750. What fraction of the original price is this?

Answer

30 people entered a raffle. 6 of these people won a prize. What fraction of people did not win a prize? Give your answer as a fraction in its simplest form.

Answer

As this question wants the number of people who did not win a prize we must first do:
30 − 6 = 24 people did not win a prize.
As a fraction this becomes which simplifies to .

Case study _unit5.19.3 Example: Estimating fractions

will not cancel.

By rounding 1347 up to 1400 and 2057 to 2100 we can cancel down the fraction to get:

Box _unit5.19.1

Note: Round to numbers that are easy to cancel, but if you round off too much, you will lose the accuracy of your answer.

7.3 Fractions of amounts

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

You then multiply that answer by the number on the top of the fraction – the numerator.

It is worth noting here that if the number on the top of the fraction is 1, multiplying the answer will not change it so there is no need for this step. Take a look at the examples below.

Case study _unit5.19.4 Example: Divide by the denominator

Method

To find of 90 we do 90 ÷ 5 = 18.

Since the number on the top of our fraction is 1, we do not need to multiply 18 by 1 as it will not change the answer.

So of 90 = 18.

Case study _unit5.19.5 Example: Multiply by the numerator

Method

To find of 42 we do 42 ÷ 7 = 6.

This means that of 42 = 6.

Since you want of 42, we then do 6 × 4 = 24.

So of 42 = 24.

Let’s go back to the jacket that used to cost £80 but is now in the sale with off. How do you find out how much it costs? Firstly, you need to find of 80. To calculate this you do:

£80 ÷ 5 = £16 and then £16 × 2 = £32

This means that you save £32 on the price of the jacket. To find out how much you pay you then need to do £80 − £32 = £48.

You will have practised finding fractions of amounts in Everyday maths 1, but have a go at the following activity to recap this important skill.

Activity _unit5.19.3 Activity 16: Finding fractions of amounts

Work out the following without using a calculator. You may double-check on a calculator if you need to and remember to check your answers against ours.

You are looking to buy house insurance and want to get the best deal. Put the following offers in order, from cheapest to most expensive, after the discount has been applied.

Table _unit5.19.1 Table 7
Company A	Company B	Company C
£120 per year	£147 per year	£104 per year
Special offer: off!	Special offer: off!	Special offer: off!

Answer

Company C is cheapest:
- of £104 = £104 ÷ 4 = £26 discount
- £104 − £26 = £78
Company A is second cheapest:
- of £120 = £120 ÷ 3 = £40 discount
  £120 − £40 = £80
Company B is most expensive:
- of £147 = £147 ÷ 7 × 2 = £42 discount
- £147 − £42 = £105

A cinema sells 2400 tickets over a weekend. They review their ticket sales and find that of the weekend ticket sales were to adults. How many adult tickets were sold?

Answer

1600 tickets sold to adults:
- 2400 ÷ 3 = 800 to give
- 2 × 800 = 1600 to give

A college has raised of its £40 000 charity fundraising target. How much money does the college need to raise to meet its target?

Answer

£16 000 needed to meet target.
40 000 ÷ 5 = 8000 to give
8000 × 3 = 24 000 to give (the amount raised)
But the question asks how much is needed to meet its target so we need to subtract the amount raised from the target:
- 40 000 − 24 000 = £16 000

Summary

In this section you have:

learned how to express a quantity of an amount in the form of a fraction
learned how to, and practised, simplifying fractions
revised your knowledge on finding fractions of amounts.

8 Percentages

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

There are different ways of working out percentages of amounts. We will take a look at the most common methods now.

Figure _unit5.20.1 Figure 9 Percentage discounts in a sale

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A series of tags sequentially reading ‘Sale up to 10%, 15%, 20%, 25%, 30%, 40%, 50%’.

Figure 9 Percentage discounts in a sale

Box _unit5.20.1

Note: You may use a different method to these ones. You may even use different methods depending on the percentage you are calculating. Do whatever works for you.

8.1 Calculating a percentage of an amount

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Method 1

Percentages are just fractions where the number on the bottom of the fraction must be 100. If you wanted to find out 15% of 80 for example, you work out of 80, which you already know how to do!

Working out the percentage of an amount requires a similar method to finding the fraction of an amount. Take a look at the examples below to increase your confidence.

Case study _unit5.20.1 Example 1: Finding 17% of 80

Method

17% of 80 = of 80, so here we do:

Extract _unit5.20.1

80 ÷ 100 = 0.8

0.8 × 17 = 13.6

Another way of thinking about this method is that you are dividing by 100 to find 1% first and then you are multiplying by whatever percentage you want to find.

Alternatively, you could multiply the value by the top number first and then divide by 100:

Extract _unit5.20.2

17 × 80 = 1360

1360 ÷ 100 = 13.6

The answer will be the same.

Case study _unit5.20.2 Example 2: Finding 3% of £52.24

Method

3% of 52.24 = of 52.24, so we do:

Extract _unit5.20.3

52.24 ÷ 100 = 0.5224.

0.5224 × 3 = £1.5672 (£ 1.57 to two d.p.)

Extract _unit5.20.4

52.24 × 3 = 156.72

156.72 ÷ 100 = £1.5672 (£1.57 to two d.p.)

Method 2

To use this method you only need to be able to work out 10% and 1% of an amount. You can then work out any other percentage from these.

Let’s just recap how to find 10% and 1%.

Box _unit5.20.2 10%

To find 10% of an amount divide by 10:

Extract _unit5.20.5

10% of £765 = 765 ÷ 10 = £76.50

10% of £34.50 = 34.50 ÷ 10 = £3.45

Hint: remember to move the decimal point one place to the left to divide by 10.

Box _unit5.20.3 1%

To find 1% of an amount divide by 100:

Extract _unit5.20.6

1% of £765 = 765 ÷ 100 = £7.65

1% of £34.50 = 34.50 ÷ 100 = £0.345 (£0.35 to two d.p.)

Hint: remember to move the decimal point two places to the left to divide by 100.

Once you know how to work out 10% and 1%, you can work out any other percentage.

Case study _unit5.20.3 Example 1: Finding 24% of 60

Find 10% first:

Extract _unit5.20.7

60 ÷ 10 = 6

10% = 6

20% is 2 lots of 10% so:

Extract _unit5.20.8

6 × 2 = 12

20% = 12

Now find 1%:

Extract _unit5.20.9

60 ÷ 100 = 0.6

4% is 4 lots of 1% so:

Extract _unit5.20.10

0.6 × 4 = 2.4

4% = 2.4

Now add the 20% and 4% together:

12 + 2.4 = 14.4

Case study _unit5.20.4 Example 2: Finding 17.5% of £328

17.5% can be broken up into 10% + 5% + 2.5%, so you need to work out each of these percentages and then add them together.

Find 10% first:

Extract _unit5.20.11

328 ÷ 10 = 32.8

10% = 32.8

5% is half of the 10% so:

Extract _unit5.20.12

32.8 ÷ 2 = 16.4

5% = 16.4

2.5% is half of the 5% so:

Extract _unit5.20.13

16.4 ÷ 2 = 8.2

2.5% = 8.2

Now add the 10%, 5% and 2.5% figures together:

Extract _unit5.20.14

32.8 + 16.4 + 8.2 = £57.40

This is a good method to do in stages when you do not have a calculator.

Box _unit5.20.4

Note: There are some other quick ways of working out certain percentages:

Extract _unit5.20.15

50% – divide the amount by two.

25% – halve and halve again.

Activity _unit5.20.1 Activity 17: Finding percentages of amounts

Use whichever method/s you prefer to calculate the answers to the following:

Find:
- a.45% of £125
- b.15% of 455 m
- c.52% of £677
- d.16% of £24.50
- e.2% of 4000 kg
- f.82% of £7.25
- g.37% of £95
The Cambria Bank pays interest at 3.5%. What is the interest on £3000?
Sure Insurance offer a 30% No Claims Bonus. How much would be saved on a premium of £345.50?
Sunshine Travel Agents charge 1.5% commission on foreign exchanges. What is the charge for changing £871?

Answer

- a.£56.25
- b.68.25 m
- c.£352.04
- d.£3.92
- e.100 kg
- f.£5.945 (£5.95 to 2 d.p.)
- g.£35.625 (£35.63 to two d.p.)
£105
£103.65
£13.065 (£13.07 to two d.p.)

Activity _unit5.20.2 Activity 18: Percentages increase and decrease

You earn £500 per month. You get a 5% pay rise.
- a.How much does your pay increase by?
- b.How much do you now earn per month?

Answer

- a.£25
- b.£525 per month.

You buy a new car for £9500. By the end of the year its value has decreased by 20%.
- a.How much has the value of the car decreased by?
- b.How much is the car worth now?

Answer

- a.The car has decreased by £1900.
- b.The car is now worth £7600.

You invest £800 in a building society account which offers fixed-rate interest at 4% per year.
- a.How much interest do you earn in one year?
- b.How much do you have in your account at the end of the year?

Answer

- a.£32 interest earned.
- b.£832 in the account at the end of the year.

Last year Julie’s car insurance was £356 per annum. This year she will pay 12% less. How much will she pay this year?

Answer

She will pay £42.72 less so her insurance will cost £313.28

A zoo membership is advertised for £135 per year. If Tracy pays for the membership in full rather than in monthly installments, she receives a 6% discount. How much will she pay if she pays in full?

Answer

She will save £8.10 so she will pay £126.90.

A museum had approximately 5.87 million visitors last year. Visitor numbers are expected to increase by 4% this year. How many visitors is the museum expecting this year?

Answer

5.87 million = 5 870 000.
4% of 5 870 000 = 234 800
5 870 000 + 234 800 = 6 104 800 people

Next you’ll look at how to express one number as a percentage of another.

8.2 Expressing one number as a percentage of another

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Sometimes you need to write one number as a percentage of another. You have already practised writing one number as a fraction of another; this just takes it a bit further.

Case study _unit5.20.5 Example 1: What percentage are women?

A class is made up of 21 women and 14 men, what percentage of the class are women?

To work this out, you start by expressing the numbers as a fraction. You then multiply by 100 to express as a percentage.

The formula is:

× 100

In this case, 21 out of a total of 35 people are women so the sum we do would be:

× 100

The fraction line is also a divide line, so if you were doing this on a calculator you would do:

21 ÷ 35 × 100 = 60%

How would you work this out without a calculator?

There are different ways you can make the calculation. Two methods are shown below.

Method 1

You start by multiplying the top number in the fraction by 100. The bottom number will stay the same:

Now you need to cancel the fraction down as much as possible:

÷ top and bottom by 5 = , then, ÷ top and bottom by 7 =

Anything over 1 is a whole number so the answer is 60.

So 60% of the class are women.

Box _unit5.20.5

15 ÷ 4 = 3.75

Method 2

The other method involves expressing the fraction as a decimal first and then converting it to a percentage. This means that you multiply by 100 at the very end of the calculation.

A local attraction sold 150 tickets last bank holiday, 102 of which were full price. What percentage of the tickets sold were at the concessionary price?

Work out the number of concessionary tickets sold:

150 – 102 = 48

Write the number of concessionary tickets sold as a fraction of the total number sold:

Cancel down your fraction:

÷ top and bottom by 6 =

Once you cannot cancel any further, you need to divide the top number by the bottom number to express as a decimal:

8 ÷ 25 = 0.32

Figure _unit5.20.2 Figure 10 Expressed as a decimal: 8 divided by 25

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8.00 ÷ 25 = 0.32. This illustration shows that the sum has been handrawn and there is a small ‘8’ above the first ‘0’ of ‘8.00’ and a small ‘5’ above the second ‘0’of 8.00.

Figure 10 Expressed as a decimal: 8 divided by 25

Finally, multiply the decimal answer by 100 to express as a percentage:

0.32 × 100 = 32%

So 32% of the tickets were sold at the concessionary price.

Activity _unit5.20.3 Activity 19: Expressing one number as a percentage of another

Use whichever method you prefer to calculate the answers to the following. Give answers to two d.p. where appropriate.

Hint: make sure your units are the same first.

What percentage:
- a.of 1 kg is 200 g?
- b.of an hour is 48 minutes?
- c.of £6 is 30p?

Answer

- a.1 kg = 1000 g
  × 100 = 20%
- b.1 hour = 60 minutes
  × 100 = 80%
- c.£1 = 100 p
  × 100 = 5%

Bea swam 50 laps of a 25 m swimming pool in a charity swim. A mile is almost 1600 m. What percentage of a mile did Bea swim?

Answer

50 × 25 = 1250 m
× 100 = 78.13% (to two d.p.)

A student gets the following results in the end of year tests:

Table _unit5.20.1 Table 8
	Maths	English	Science	Art
Mark achieved	64	14	72	56
Possible total mark	80	20	120	70

Calculate her percentage mark for each subject.

Answer

Maths: × 100 = 80%

English: × 100 = 70%

Science: × 100 = 60%

Art: × 100 = 80%

Susan is planting her flower beds. She plants 13 yellow flowers, 18 white flowers and 9 red ones. What percentage of her flowers will not be white?

Answer

Number not white = 13 + 9 = 22
Total number she is planting = 13 + 18 + 9 = 40
× 100 = 55%
55% of the flowers will not be white.

A building society charges £84 interest on a loan of £1200 over a year. What percentage interest is this?

Answer

× 100 = 7%
The interest rate is 7%.

Next you will look at percentage change. This can be useful for working out the percentage profit (or loss) or finding out by what percentage an item has increased or decreased in value.

8.3 Percentage change

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Watch the video below on how to calculate percentage change, then complete Activity 20.

Download this video clip.Video player: bltl_1_5_2_percentage.mp4

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Transcript

Next, divide this difference by the original cost, and then multiply by 100. £3.60 divided by £24, x 100, = 15. The cost of the train ticket has increased by 15%.

End transcript

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Activity _unit5.20.4 Activity 20: Percentage change formula

Practise using the percentage change formula which you learned about in the video above on the four questions below. Where rounding is required, give your answer to two decimal places.

Last year your season ticket for the train cost £1300. This year the cost has risen to £1450. What is the percentage increase?

Answer

Difference: £1450 − £1300 = £150
- Original: £1300
- Percentage change = × 100
- Percentage change = 0.11538... × 100 = 11.54% increase (rounded to two d.p.)

You bought your house 10 years ago for £155 000. You are able to sell your house for £180 000. What is the percentage increase the house has made?

Answer

Difference: £180 000 − £155 000 = £25 000
- Original: £155 000
- Percentage change = × 100
- Percentage change = 0.16129... × 100 = 16.13% increase (rounded to two d.p.)

You purchased your car 3 years ago for £4200. You sell it to a buyer for £3600. What is the percentage decrease of the car?

Answer

Difference: £4200 − £3600 = £600
- Original: £4200
- Percentage change = × 100
- Percentage change = 0.14285... × 100 = 14.29% decrease (rounded to two d.p.)

Stuart buys a new car for £24 650. He sells it 1 year later for £20 000. What is the percentage loss?

Answer

Difference: £24 650 − £20 000 = £4650
Original: £24 650
Percentage change = × 100
4650 ÷ 24 650 × 100 = 18.86% loss (rounded to two d.p.)

Summary

In this section you have:

found percentages of amounts
calculated percentage increase and decrease
calculated percentage change using a formula
expressed one number as a percentage of another.

9 Fractions, decimals and percentages

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Figure _unit5.21.1 Figure 11 Equivalent decimals, fractions and percentages

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Illustration showing equivalences: Decimal, 0.5; Fraction, 1/2; Percentage, 50%.

Figure 11 Equivalent decimals, fractions and percentages

9.1 Converting between percentages, decimals and fractions

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Download this video clip.Video player: bltl_1_6_fractions_decimals_percentages.mp4

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Transcript

What about 25%? Can you work out the equivalent decimal of 25% before the answer is revealed? Remember that 'cent' means 100.

To turn 25% into its equivalent decimal, you need to divide 25 by 100, which = 0.25.

Now looking at the table, you can see how percentages, decimals and fractions relate to each other.

End transcript

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Lets look in more detail at changing a percentage to a fraction.

Case study _unit5.21.1 Example: 50% is

As you can see this percentage is essentially a fraction of 100. However you can simplify it to .

To change a percentage to a fraction, put the percentage over 100 and simplify if possible.

Sometimes we might see a percentage like this: 12.5%.

If we use the method above we get but we can’t have a decimal in a fraction.

Extract _unit5.21.1 Method 1: × 10

× top and bottom by 10 = =

Extract _unit5.21.2 Method 2: × 2

× top and bottom by 2 = =

Activity _unit5.21.1 Activity 21: Converting between percentages, decimals and fractions

Express these percentages as decimals:
- a.62%
- b.50%
- c.5%
Express these decimals as percentages:
- a.0.02
- b.0.2
- c.0.752
- d.0.055
Express these percentages as fractions:
- a.15%
- b.2.5%
- c.37.5%

Answer

- a.0.62
- b.0.5
- c.0.05
- a.2%
- b.20%
- c.75.2%
- d.5.5%
- a. =
- b. × top and bottom by 10 = =
- c. × top and bottom by 10 = =
  
  You may have multiplied by different numbers to get rid of the decimal in the last two questions. However, your final answers should still be the same as ours.

Now have a go at matching these fractions to decimals and percentages.

Activity _unit5.21.2 Activity 22: Matching fractions, decimals and percentages

Choose the correct fraction for each percentage and decimal.

Using the following two lists, match each numbered item with the correct letter.

a. 62.5% = 0.625 =
b. 40% = 0.4 =
c. 8% = 0.08 =
d. 35% = 0.35 =

The correct answers are:

1 = d
2 = b
3 = c
4 = a

Next you’ll look in more detail at how to change a fraction to a percentage.

9.2 Changing a fraction to a percentage

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

There are two ways you can do this.

Method 1

To change a fraction into a percentage, multiply it by (essentially, you are just multiplying the top number by 100 and the bottom number will stay the same).

Case study _unit5.21.2 Example: Change into a percentage

Extract _unit5.21.3

× =

This cancels to = 75%

Box _unit5.21.1

Note: Remember anything over 1 is a whole number. If you do not end up with a 1 on the bottom, you will have to divide the top number by the bottom one to get your final answer.

Method 2

Divide the top of the fraction by the bottom (to express the fraction as a decimal) and then multiply the answer by 100.

Case study _unit5.21.3 Example: = 3 ÷ 4 = 0.75

Figure _unit5.21.2 Figure 12 Expressed as a decimal: 3 ÷ 4

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3.00 ÷ 4 = 0.75. This illustration shows that the sum has been handrawn and there is a small ‘3’ above the first ‘0’ of ‘3.00’ and a small ‘2’ above the second ‘0’of 3.00.

Figure 12 Expressed as a decimal: 3 ÷ 4

0.75 × 100 = 75%

Activity _unit5.21.3 Activity 22: Changing a fraction to a percentage

Express these fractions as percentages:
- a.
- b.
- c.

Answer

- a.37.5%
- b.90%
- c.80%

Now you’ll look at changing a fraction to a decimal.

9.3 Changing a fraction to a decimal

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Again there are two ways to do this, both based on the two methods just shown for changing a fraction to a percentage.

Method 1

Case study _unit5.21.4 Example: Change the fraction into a percentage and divide by 100

× = which cancels to = 25%

Now convert to a decimal by dividing by 100:

25 ÷ 100 = 0.25

Method 2

Case study _unit5.21.5 Example: Divide the top of the fraction by the bottom

= 1 ÷ 4 = 0.25.

Figure _unit5.21.3 Figure 13 Expressed as a decimal: 1 ÷ 4

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1.00 ÷ 4 = 0.25. This illustration shows that the sum has been handrawn and there is a small ‘1’ above the first ‘0’ of ‘1.00’ and a small ‘2’ above the second ‘0’of 1.00.

Figure 13 Expressed as a decimal: 1 ÷ 4

Activity _unit5.21.4 Activity 23: Changing a fraction to a decimal

Express these fractions as decimals:

Answer

0.4
0.125
0.3

Summary

In this section you have:

learned about the relationship between fractions, decimals and percentages and are now able to convert between the three.

10 Ratio

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

As you can see from Figure 14, ratio is an important part of everyday life.

Figure _unit5.22.1 Figure 14 Day-to-day ratio

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Cartoon of a teacher and several children playing in a school. A sign reads ‘Adult to child ratio must be 1:4’.

Figure 14 Day-to-day ratio

The order of the ratio is very important. Consider the following:

Extract _unit5.22.1

Julia attends a drama club where 100 members are men and 150 members are women. What is the ratio of women to men at the drama club?

The ratio of women to men is 150:100

If you were asked for the ratio of men to women it would be 100:150

10.1 Simplifying ratios

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

Sometimes you need to work out the ratio from the quantities you have.

Let’s look at another example.

Case study _unit5.22.1 Example: Recipes and ratio

Look at this recipe for a mocktail:

Sunset Smoothie

50 ml grenadine
100 ml orange juice
150 ml lemonade

The ratio of the ingredients is:

grenadine:orange juice:lemonade
50 : 100 : 150

To simplify this ratio you can divide all of the numbers by 50 (or by 10 and then 5).

This gives the ratio of grenadine to orange juice to lemonade as 1:2:3.

Activity _unit5.22.1 Activity 24: Simplifying ratios

Simplify the following ratios:

The ratio of women to men in a class is 15:20.
The ratio of management to staff in a warehouse is 10:250.
The ratio of home to away supporters is 24 000 to 8000.
The ratio of votes in a local election was candidate A 1600, candidate B 800, Candidate C 1200.
The ratio of fruit in a bag of mixed dried fruit is 150 g currants, 100 g raisins, 200 g sultanas and 50 g mixed peel.

Answer

Women to men is 3:4 (divide both sides by 5).
Management to staff is 1:25 (divide both sides by 10).
Home to away supporters is 3:1 (divide both sides by 8000 or by 1000 and then by 8).
A to B to C is 4:2:3 (divide each part of the ratio by 400 or by 100 and then by 4).
Currants to raisins to sultanas to mixed peel is 3:2:4:1 (divide by 50 or by 10 and then 5).

Ratio questions can be asked in different ways. There are three main ways of asking a ratio question. Take a look at an example of each below and see if you can identify the differences.

Case study _unit5.22.2 Type 1

A recipe for bread says that flour and water must be used in the ratio 5:3. If you wish to make 500 g of bread, how much flour should you use?

Case study _unit5.22.3 Type 2

You are growing tomatoes. The instructions on the tomato feed say ‘Use 1 part feed to 4 parts water’. If you use 600 ml of water, how much tomato feed should you use?

Case study _unit5.22.4 Type 3

Ishmal and Ailia have shared some money in the ratio 3:7. Ailia receives £20 more than Ishmal. How much does Ishmal receive?

10.2 Solving ratio problems where the total is given

The Open University — Tue, 01 Oct 2019 13:34:23 GMT

The best way for you to understand how to solve these problems is to look through the worked example in the video below.

Download this video clip.Video player: bltl_1_7_1_ratio.mp4

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Transcript

To understand how to solve ratio problems where the total is given, it's best to look through a worked example.

An art shop is ordering paint. They work out that they need to order red and blue paint in the ratio 3:4. They order a total of 56 cans of paint. How many of those cans are blue paint?

Now, you know that one part of paint is 8 cans, you can work out what four parts are worth. 8 x 4 = 32. So the art shop needs to order 32 cans of blue paint.

The total of red paint and blue paint should be 56. So you can check your answer by adding together the number of cans of each. 32 + 24 is indeed 56 cans.

So in summary, the steps for solving a ratio question like this are:

One. Add together the parts of the ratio. 3 + 4 = 7.

Two. Take the total amount given and divide by the sum of the ratio parts. 56 divided by 7 = 8.

Three. Finally, take the answer for step two, 8, and multiply by whichever part of the ratio you're interested in finding. 8 x 4 = 32.

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Activity _unit5.22.2 Activity 25: Ratio problems where the total is known

Try solving these ratio problems:

To make mortar you need to mix soft sand and cement in the ratio 4:1. You need to make a total of 1500 g of mortar.
How much soft sand will you need?

Answer

Add the parts of the ratio:
- 4 + 1 = 5
Divide the total amount required by the sum of the parts of the ratio:
- 1500 g ÷ 5 = 300 g
Since soft sand is 4 parts, we do 300 g × 4 = 1200 g of soft sand.
Check by working out how much cement you need. Cement is 1 part so you would need 300 g:
- 1200 g + 300 g = 1500 g which is the correct total.

To make the mocktail ‘Sea Breeze’ you need to mix cranberry juice and grapefruit juice in the ratio 4:2.
You want to make a total of 2700 ml of mocktail. How much grapefruit juice should you use?

Answer

Add the parts of the ratio:
- 4 + 2 = 6
Divide the total amount required by the sum of the parts of the ratio:
- 2700 ml ÷ 6 = 450 ml
Since grapefruit juice is 2 parts, we do 450 ml × 2 = 900 ml of grapefruit juice.
Check by working out how much cranberry juice you would use:
- 4 × 450 = 1800
  1800 ml + 900 ml = 2700 ml
  You may have simplified the ratio to 2:1 before doing the calculation, but you will see that your answers are the same as ours.

The instructions to mix Misty Morning paint are mix 150 ml of azure with 100 ml of light grey and 250 ml of white paint.
How much light grey paint would you need to make 5 litres of Misty Morning?

Answer

Start by expressing and then simplifying the ratio:
- 150:100:250 which simplifies to 3:2:5 = 10 parts
- 5 litres = 5000 ml (converting to ml makes your calculation easier.)
Divide the total amount required by the sum of the parts of the ratio:
- 5000 ÷ 10 = 500 so 1 part = 500 ml
- Light grey is 2 parts:
  2 × 500 = 1000 ml or 1 litre
Check:
- azure is 3 parts: 3 × 500 = 1500 ml or 1.5 litres
- white is 5 parts: 5 × 500 = 2500 ml or 2.5 litres
- 1000 + 1500 + 2500 = 5000 ml or 5 litres

You want to make 14 litres of squash for a children’s party. The concentrate label says mix with water in the ratio of 2:5.
How much concentrate will you use?

Answer

Add the parts of the ratio:
- 2 + 5 = 7
Divide the total amount required by the sum of the parts of the ratio:
- 14 litres ÷ 7 = 2 litres so 1 part = 2 litres
(Note: this calculation was straightforward so there was no need to convert to ml.)
Since the concentrate is 2 parts you will need 2 litres × 2 = 4 litres of concentrate.
Check:
- Water is 5 parts:
- 5 × 2 litres = 10 litres
- 4 + 10 = 14 litres.

A man leaves £8400 in his will to be split between 3 charities:
- Dogs Trust, RNLI and MacMillan Research in the ratio 3:2:1.
How much will each charity receive?

Answer

Add the parts of the ratio:
- 3 + 2 + 1 = 6
Divide the total amount required by the sum of the parts of the ratio:
- £8400 ÷ 6 = 1400
- – The Dogs Trust receives 3 parts: 3 × £1400 = £4200
- – The RNLI receives 2 parts: 2 × £1400 = £2800
- – MacMillan Research receives 1 part so: £1400
- Check:
  4200 + 2800 + 1400 = £8400

Next you’ll look at ratio problems where the total of one part of the ratio is known.