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# Unit 8: Relationships Among Numbers

## 8 Introduction

In this unit, we will look at other ways numbers can be expressed. Whether you are shopping around for the best deal, or reading an article online, understanding percentages and ratios can be useful in your everyday life.

### Practice Quiz

Check your understanding of percentages and related calculator skills before you start, by giving the Unit 8 pre quiz a try, then use the feedback to help you plan your study.

The quiz does not check all the topics in the unit, but it should give you some idea of the areas you may need to spend most time on. Remember, it doesn’t matter if you get some or even all of the questions wrong—it just indicates how much time you may need for this unit!

### 8.0.1 What to Expect in this Unit

This unit should take around ten hours to complete. In this unit you will learn about:

• Understanding percentages.
• How percentages are related to fractions and decimals.
• Calculating with percentages by hand and with the calculator.
• Ratios.
• Percentages and ratios in everyday life.

In Section 1, percentages and related calculator skills will be explored. Depending on your comfort level, this section might require you to spend extra time. Sale prices, restaurant tipping, and online shopping are a few of the examples you will look at.

In Section 2, you will study ratios. Hopefully, you will find the content of this section to be familiar as you apply ratios to decide whether you can afford to buy a hot tub.

In Section 3, you will have the opportunity explore these topics at a deeper level. Percent change and the Golden Ratio are two of the topics you will discover.

Section 4 provides a self-check, so that you can consider the development of your skills. Remember that your confidence will continue to increase the more you practice. You can do this by working through exercises on purchasing a flat-screen television and baking cookies!

## 8.1 Percentages

### Introduction

Articles you read and newscasts you may watch or listen to often mention the results of surveys and studies in percents. Literally, percent means per one hundred. So 25% (which is read as “twenty-five percent”) means “25 out of one hundred.”

One of the reasons percentages are useful is that you can compare them easily. For example, if a restaurant noted that 42% of customers opted for a meat dish, 35% chose a vegetarian dish, and 23% ordered fish, you can quickly conclude that the meat dish was most popular.

Note: The preferences of the customers in the example above will add up to 100; in this case .

### 8.1.1 Writing a Percentage as a Fraction or Decimal

Fractions, decimals, and percentages are all interchangeable, so you can choose to use whichever is most appropriate for your situation. From the definition of a percentage, 75% means , which can, as you know, be reduced to . Alternatively, means , or 0.75. So, .

You can see from the pizzas below that dividing them into and , or 25% and 75%, 0.25 and 0.75 always amounts to the same thing.

#### Summary

To change a percentage to a fraction or decimal, interpret the percent symbol to mean “divided by 100.” You then have a choice to display your result as a fraction or decimal.

Since dividing a number by 100 has the same effect as moving its decimal point two places to the left, you have a valuable shortcut when you turn a percent into a decimal. [ Looking for the decimal point in a whole number? Remember, the decimal point can be inserted at the end of your number behind the ones place. ]

For example: 6% = .

### 8.1.2 Writing a Decimal as a Percentage

To turn a decimal back into a percentage, you can move the decimal point two places in the opposite direction to the right.

As an example: .

Let’s take a closer look—“percent” means “over 100,” so 100% is actually an equivalent way of expressing , which is the number 1. Multiplying by 100% does not change the value of a number; it just makes it look different.

#### Example

If you’d like, you can show the conversion with one or both of the following steps:

For example: .

### 8.1.3 Writing a Fraction as a Percentage

Using the same idea that multiplication by 100% means multiplication by 1, you can also convert fractions into percentages.

#### Example

See this additional example turning a fraction into a percentage (click on “View document”).

View document

As an alternative, you can turn the fraction into a decimal first, and then move the decimal point:

Example: .

#### Summary

To change a decimal or fraction to a percentage, multiply by 100%, or change to a decimal and then to percent.

There are other alternative ways, such as multiplying the numerator and denominator of the fraction to be converted by , shown in this example:

For this last example, note that the fraction can be expanded by multiplication so that the denominator is turned into 100. For some fractions, this gives you another alternative for the conversion. For this last example, it looks like this:

Let’s revisit the different ways in which you can convert decimals and fractions into percents (Click on “View document”).

View document

### 8.1.4 The Importance of the Percent Sign

A mistake that many people make is that they think of % as a unit, instead as a symbol that actually affects the number itself. 28% is not equal to 28, it is actually, or 0.28.

This example summarized: , not 28.

### 8.1.5 Converting

#### Activity: Fractions, Decimals, and Percentages

[ Want to get more comfortable with converting among fractions, decimals, and percentages? Play this game! ] (a) In your math notebook, turn each of the given percentages into a decimal as well as into a fraction in lowest terms.

• (i) 10%
• (ii) 25%
• (iii) 50%
• (iv) 125%
• (v) 0.5%
##### Comment

To turn a percentage into a decimal, how far and in which direction should you move the decimal? To turn the percentage into a fraction, which number do you divide by? Don’t forget to reduce the fraction if possible.

(a)

• (i) Decimal: Fraction:
• (ii) Decimal: Fraction:
• (iii) Decimal: Fraction:
• (iv) Decimal: Fraction:
• (v) Decimal: Fraction:

(b) Write each of the following given numbers as percentages.

• (i)
• (ii)
• (iii)
• (iv) 0.4
• (v) 0.0075
• (vi) 1.2
##### Comment

There are several different ways of doing this. Do you like to go through a decimal? Do you prefer to multiply by 100%? (This last approach will be shown in the solutions.)

(b) Multiplying each fraction or decimal by 100% gives the following answers:

• (i)
• (ii)
• (iii)
• (iv)
• (v)
• (vi)

By no means did you have to do it this way, but don’t forget that your answer must include a percent symbol.

### 8.1.6 Calculator Exploration: Percentage Conversions

The calculator can be accessed on the left-hand side bar under Toolkit.

In this exploration, you will use the calculator for percentage conversions.

#### To Write a Percentage as a Fraction or a Decimal

Remember that the percent sign means “out of one hundred.” So, 5% means , and this is how it is entered on the calculator.

Enter 5% as on the calculator, and click the equals sign, or hit Enter on your keyboard.

The key sequence is

The calculator gives the fraction in its reduced form with the decimal equivalent.

##### Activity: Converting Percentages to Fractions

Try writing these percentages as fractions and as decimals.

(a) 88%

###### Comment

Clear the last calculation before you start, and remember that % means “out of one hundred.”

(a) Entering 88% as 88 divided by 100 gives 0.88, so .

(b) 250%

(b)

#### To Write a Decimal Number as a Percentage

To write a decimal number as a percentage, multiply the number by 100%. The calculation is quite easy to do without the calculator because you can move the decimal point two places to the right. For example, 0.8 becomes . Let’s check that the calculator does the same. Enter the calculation . The calculator gives the answer 80.

Remember that you must include the % sign when you quote the answer.

##### Activity: Converting Decimals to Percentages

Use the calculator to write these decimal numbers as percentages:

(a) 0.28

(a) To convert 0.28 to a percentage, multiply by 100%. Using the calculator, multiply by 100, and then attach the % sign when you write the answer down, as follows: .

Note: We do not use the percent key when converting a decimal to a percentage. When you use the percent key, it is converting the value to its decimal equivalent. Here, you are just trying to find the value that belongs in front of the percent sign.

(b) 1.052

(b) . (Did you remember to attach the % sign when you wrote the answer down?)

(c) 0.006

(c) .

### 8.1.7 To Write a Fraction as a Percentage

You can use the same method, multiplying by 100%, to write a fraction as a percentage. For example, enter into the calculator, then multiply by 100. It’s useful, though not essential, to put parentheses around the fraction. The calculator shows , or 12.5. Therefore, .

#### Activity: Making Percentages From Fractions

Use the calculator to write these fractions as percentages, giving an exact answer followed by a decimal, rounded to 2 decimal places, where necessary.

(a)

##### Comment

Enter the fraction, then multiply by 100. Remember to attach the percent sign when you give your answer.

(a) , or 83.33%.

Note: Remember, do not use the percent key when converting a fraction to a percentage.

(b)

(b) , or 0.25%.

(c)

(c) , or 157.14%

#### The Percent Key on the Calculator

You may have noticed that there is a percent key on the calculator in the top row of keys below the number pad. There are some calculations where using this key will save you time, but be careful, as the calculator does not always behave how you may expect it to.

Enter on the calculator. The calculator converts 5% to a fraction, but does not give the decimal equivalent.

So the calculator can help you convert a percentage to a fraction, but it does not change the fraction to a decimal. Also, the calculator does not help when you convert a decimal or a fraction into a percentage. For these reasons, it is better to avoid using the % key on the calculator for conversions, and to divide or multiply by 100 instead.

##### To recap:
• To write a percentage as a fraction or a decimal, write the percentage as a number divided by 100 and click on the equals sign, or hit Enter on the keyboard.
• To write a decimal number or a fraction as a percentage, multiply by 100 on the calculator and then attach the % sign when you give your answer.
• Entering a percentage followed by the percent key and clicking equals gives the percentage as a fraction only.

### 8.1.8 Find a Percentage of a Number

Suppose a mathematics exam is out of 75 points, and students need to achieve at least 60% to pass. How many points guarantee a passing grade?

To pass, the student has to earn at least 60% of 75. Here, we can change the percentage into a fraction, then calculate to find the required point amount. You can work out the points necessary by reducing the fraction and canceling further.

Here is one way of reaching the answer:.

Or, you can multiply 60 by 75, then divide by one hundred:. Both ways show that the student needs 45 points.

#### Summary

To find a given percentage of a number, change the percentage to a fraction (or decimal) and multiply by the number.

### 8.1.9 Going Solo

#### Activity: Going Solo

(a) A radio program reported the following:

“Our ‘Going Solo’ survey of 4,000 single people found that only 1 in 5 is happy on their own.”

Write “1 in 5” as a fraction, a decimal, and a percentage. Which form do you think is easiest to understand?

##### Comment

Which numbers create the fraction? How does this fraction convert to a percent? How does it convert to a decimal?

(a) “1 in 5” can be written as , or , as well as .

The report could have said “one-fifth” or “20%” of people in the survey, but “1 in 5” makes the proportion easy to visualize for people not comfortable with fractions and percentages. Which form is easiest to understand depends on your way of thinking.

Remember, all of these express the same number, although “0.2 of the people” would probably not be used, because it’s much harder to visualize and put in context. In part (b) you will determine how many people this is.

(b) How many of the 4,000 “Going Solo” survey participants seem to be content with being single?

##### Comment

You know that it’s one-fifth of the group, and remember, “of” means “multiply.”

(b) . So, 800 out of the 4,000 surveyed singles feel happy on their own.

### 8.1.10 Which Percent is It?

You probably agree that comparing percentages can be easier than comparing fractions. When we have a certain quantity, how do we write it as a percent of another, usually larger, number?

For example, if 42 people out of a group of 70 people agree to participate in a new community project, what percentage of the group is this?

Whenever you have a part of a group and the whole of the entire group, you can calculate percent by setting up the ratio of “part over whole.” You can then turn this fraction successfully into a percentage.

So in this case, turn into a percent, first by multiplying by 100% and then by reducing by 10 and then by 7:

(Note: You also could have reduced to first and then turned it into a percent.)

These calculations show us that 60% of the group agree to participate in the community project.

As you know, the calculation can be performed even quicker with a calculator, as can the other percent calculations.

#### Summary

To express one number as a percentage of another, divide the first number by the second and convert the result into percent by multiplying by 100%.

### 8.1.11 Calculator Exploration: Using Percentages

The calculator can be accessed on the left-hand side bar under Toolkit.

In the following exploration, you will use the calculator for calculations involving percentages.

#### Finding a Percentage of a Number

Suppose that residents in a town are asked their views on a proposed development of a wind farm near the town. The local paper says that 450 people voted, with 54% in favor of the proposal, 36% against the proposal and 10% saying that they were undecided. How many people were in favor of the proposal?

To find the number of people who were in favor, you need to find 54% of 450. The calculation is: .

Enter this into the calculator and you will see:

So, 243 people were in favor of the proposal.

The percent key is useful here. Enter .

The calculator shows:

Again, you can see that 243 people were in favor of the proposal.

##### Activity: Methods for Finding Percentages

Now it’s your turn. Try both methods (first, entering the percentage as a fraction, and second, using the percent key) to find the number of people who were against the proposal. Then find the number of people who were undecided about the proposal. (Try this calculation without using the calculator.)

###### Comment

For the first method, enter the percentage as a fraction, then multiply by 450. For the second method, enter the percentage followed by the percent key, then multiply by 450.

Since 36% of the 450 people were against the proposal, you want to calculate 36% of 450 to find the exact number. Here are the two methods on the calculator.

So, 162 people were against the proposal.

You know that 10% of the 450 people were undecided, so you need to find 10% of 450. You should be able to do this calculation without the calculator: 10% is one-tenth, and one-tenth of 450 is 45. So, 45 people were undecided. The calculator should give you the same answer, but do watch for easy calculations that you can do in your head—it’s good practice!

As a check, the three numbers you have calculated should add to 450. The number in favor was 243, the number against was 162 and the number undecided was 45, and these do add to 450.

### 8.1.12 To Write One Number as a Percentage of Another

[ Source: U.S. Census (state data, “Search”; national data, “Map”) ]

#### Activity: Writing One Number as a Percentage of Another

(a) In 2010, the population of Maryland was about 5,774,000, and the total population of the United States was about 308,746,000. What was the population of Maryland as a percentage of the total population of the United States?

##### Comment

To find the percentage, you take the ratio of “part to whole” and multiply by 100%.

(a) The calculation required is .

Try this on the calculator. The calculator should show:

You need to round the answer. How many decimal places would make sense here?

This rather depends on who the answer is for. Probably, either one or two decimal places are sufficient. Round the answer to 2 decimal places.

Rounded to 2 decimal places, the population of Maryland was 1.87% of the total population of the United States in 2010.

(b) California has the largest population of any state in the US, about 37,254,000 in 2010 and the total population of the United States was about 308,746,000. Find the population of California as a percentage of the total population of the United States, rounding your answer to 2 decimal places.

##### Comment

Write the part divided by the whole, then multiply by 100%.

(b) The calculator shows:

The population of California, rounded to 2 decimal places, was about 12.07% of the total population of the United States in 2010.

#### To recap:

• To find a percentage of a number, write the percentage as a fraction (by dividing by 100) and multiply by the number. Or use the percent key, by entering the percentage followed by the percent key and multiply by the number.
• To write one number as a percentage of another, write the part divided by the whole and multiply by 100%.

References

U.S. Census Bureau. “2010 Census Interactive Population Map.” U.S. Census Bureau. http://2010.census.gov/ 2010census/ popmap/ index.php

U.S. Census Bureau. “2010 Census Interactive Population Search.” U.S. Census Bureau. http://2010.census.gov/ 2010census/ popmap/ ipmtext.php

### 8.1.13 World Populations

This activity will require the use of a calculator.

The calculator can be accessed on the left-hand side bar under Toolkit.

#### Activity: Different Nationalities

In 2011 it was estimated that 1 in 6 of the world’s population were living in India and 1 in 23 were living in the USA.

Convert these values to percentages using your calculator to make them easier to compare.

1 in 6 is the fraction . In percentages, this becomes .

1 in 23 is the fraction . In percentages, this becomes .

Note here that we have rounding to the nearest whole number as population figures are necessarily estimates.

### 8.1.14 Money

#### Activity: Where Does the Money Go?

Let’s suppose your gross monthly income (that is, your income before taxes) is $3,960. Financial experts recommend that no more than$1,386 of this amount should go toward your mortgage. Based on your monthly salary, you should also spend no more than $316.80 per month on car payments. Additionally, it is advisable to place$792 of your gross monthly income into a savings account.

If you are putting these exact amounts toward these categories each month, which percentage of your gross monthly income will be spent on the following expenses?

(a) Mortgage

Did you try rewording the given information, then translating it? For example, $1,386 out of$3,960 should go toward mortgage payments. This translates into the quotient . The $symbols cancel (because they are units) and the quotient can be worked out. Remember, you will need to convert your unit-less decimal to a percentage. ##### Answer (a) The percentage spent on the mortgage is . (b) Car Payments ##### Answer (b) The percentage spent on car payments is . (c) Savings ##### Answer (c) The percentage put into savings is . Of course, these numbers represent general advice. It might be better to purchase a used vehicle that is not too old and has a reasonable number of miles on it, and try to pay for it outright from your savings account, rather than to have a car payment. ### 8.1.15 Saving for Retirement #### Activity: Saving for Retirement You know that saving for retirement is important, and there are many resources available to tell you how much you should save per paycheck. [ For many employed people in the U.S., participating in a company’s 401(k) plan with a company match or setting up a Roth IRA with a broker firm is a good first step. ] Do you know which percentages of pretax income (gross income) many reputable financial advisors recommend to put away for retirement if a client starts to save for retirement in his or her: (a) Twenties? ##### Comment According to financial advisors, (a) people who start in their twenties should put 15% of their pretax income toward retirement savings, (b) Thirties? ##### Comment (b) people who start in their thirties should save 25% of their pretax dollars for retirement (c) Forties? ##### Comment (c) and people who start in their forties need to put away 35%. [ Discouraged? Good news: Saving a little is better than not saving at all. And if you work in a job you enjoy, you probably won’t mind working until later in life, which means your savings have a longer time to grow and you won’t rely on them as early. ] Starting later than mid 40s is even more of a challenge as these recommended percentages increase further. According to this advice, how much should the following people, who are just starting to create their first retirement savings, put into their respective retirement accounts each year? (All yearly incomes given are pretax.) (i) A 36-year-old groundskeeper earning$25,000 a year.

(i) The groundskeeper is in his or her thirties, and thus should save per year. (This would be or about $520 a month.) (ii) A 43-year-old nurse earning$70,000 a year.

(ii) The nurse is in his or her forties, and thus should save per year. (This would be about $2040 a month.) (iii) A 27-year-old entrepreneur earning$46,000 a year.

(iii) The entrepreneur is in his or her twenties, and thus should save per year. (This would be about $575 a month.) Looking at these percentages, you see why it is a good idea to start paying into a retirement plan or account as soon as you can. ### 8.1.16 Sales Tax The calculator can be accessed on the left-hand side bar under Toolkit. Percentages are used frequently in business transactions when calculating extra charges such as sales tax, a tip left in a restaurant, working out discounts on sale items, or specifying interest on a loan or savings account. As you know, sales tax and discounts are given in percent. To calculate the total price of a purchase, you can work out the extra charge (or discount) and then add it to (or subtract it from) the original price. As an example, the sales tax in Oklahoma, as of 2011, was 4.5%. To work out the total price on a$148 lounge chair purchased there, you can calculate 4.5% of $148, then add that amount to the ticket price. If you just want a rough idea what the item will cost at check out, an estimation will give you an idea. Let's make an estimate, first. 4.5% is close to 5%, and$148 is approximately $150. You can find 10% of a number very easily by dividing the amount by 10, and 5% is half of 10%, so taking half of the amount from the previous step will do the trick. This makes the added tax about and the entire purchase approximately . You also know that you rounded both figures up—the original price, as well as the sales tax—so if you have at least this amount to pay with, it will be more than enough to make this lounge chair yours. Using the calculator and working exactly, 4.5% of$148 is .

This is reasonably close to the estimate we found above for the sales tax, which was $7.50. The total price of for this lounge chair including Oklahoma sales tax is . Note: You can also calculate if you don’t want to do the calculation in two steps. The 1.045 then stands for , because you are paying 100% of the price for the chair, plus 4.5% of the price additional in tax. Calculating 104.5% of the original cost, therefore, gives the final price. ### 8.1.17 Calculating Sales Tax #### Activity: Expensive Sales Tax (a) Sabrina was visiting California, and bought a pair of sneakers with a ticket price of$69. The California sales tax rate at the time was 8.25%. First, without using a calculator, give an estimate of what the shoes cost in all. Afterward, find the exact price of her purchase.

The calculator can be accessed on the left-hand side bar under Toolkit.

It does not matter if you figure the sales tax as an estimated 10% (rounded up very roughly) or 8% (rounded down to the closest percent). Working with 10% clearly is much easier. The $69 rounds to$70, and since 10% of $70 is$7 (just divide 70 by 10), the total comes out to less than $77. Now, use the exact numbers to find the actual purchase price Sabrina paid. Pay close attention to your decimal places, as you are converting a one-digit percentage to a decimal. ##### Answer (a) 8.25% of$69 is (rounded to the closest cent) The total price was .

You may also have calculated it like this:.

(b) [ Sabrina was fortunate. If you travel to England now, 20% VAT is added to all purchases. ] When Sabrina traveled to Europe in 2010, she was amazed to see that England has a 17.5% sales tax, which is called VAT (value added tax). In London, she bought a pair of boots that were £26 before tax. After calculating a rough estimate of sales tax and final purchase amount, can you find the exact sales tax in your head using mental strategies from chapter 2? Then find the exact purchase amount.

##### Comment

Make an estimate first. 20% of £30 is approximately . Since we rounded up, we expect the VAT to be less than £6.

For the mental approach, think about breaking the VAT of 17.5% down into . Do you see that each of these partial percentages is half of the last? You also know how to find 10% of an amount. Use these strategies to do your mental math.

(b) 10% of £26 is £2.60.

5% of £26 is £1.30 (half of £2.60).

2.5% of £26 is £0.65 (half of £1.30).

Therefore, 17.5% of £26 is , which is less than £6, and thus reasonable when compared to our estimate.

The final price is the original cost plus the VAT is .

The corresponding amount in U.S. dollars depends on the current exchange rate. In 2010, the exchange rate was about $1.60 per British pound. This would make the boots approximately £. We’ll look at more conversions like this in Unit 9. ### 8.1.18 Leaving a Tip If you go out to eat, you know that it is customary to leave a tip (gratuity) for the server based on the amount of your bill. 15% is considered typical, although many people agree that if you enjoyed the service, a tip of 20% is appropriate. [ Did you know that most U.S. wait staff are paid less than minimum wage, and that the main portion of their income comes from tips? They are also required to pay income tax on the gratuity they receive. ] How do you figure out how much to give as a tip? This video helps you to learn how to calculate a standard 15% tip quickly: Interactive feature not available in single page view (see it in standard view). Watch the video and then review your knowledge in the next activity. #### Activity: Leaving a Tip Imagine you are at a restaurant, and you pay your bill with your debit card. You decide to tip about 15% on the$41.68 bill (which already included sales tax).

How much tip will you add, and what is the total cost of the meal with tip?

$41.68 can be rounded up to$42.00. Dividing by 10 gives you 10% of $42.00, which is$4.20. Multiplying by 2 creates 20% being $8.40. Did you catch what he did next? He is taking the average of the sum of these two dollar amounts. He can do this because the average of 10% and 20% is . Now do it with the money: . Lastly, . The standard tip (15%) is about$6.30, and the total amount would be about $48.30. In such a situation, you probably will consider to round up to$49.00 or even $50.00 if the service was good (the latter coming close to a more generous 20% in gratuity). If the averaging arithmetic is a little tricky, you can round the sums before doing it. For example, round$4.20 to $4 and$8.40 to $8. It’s easier to see that the average of those is , which is still reasonable for a tip. ### 8.1.19 Online Shopping #### Activity: Online Shopping A website that you sometimes shop on offers free shipping and no sales tax. This week, they are also having a promotion which will make your entire order 20% off. You place an order for items that without the discount would cost$45.50. How much will you pay?

How much is the 20% discount on the $45.50 order? You can use it to determine the final cost. ##### Answer How about figuring out the discount mentally? 10% of$45.50 is $4.55. Double that to 20%, which is . You could also calculate the discount directly, maybe with a calculator: 20% of$45.50 is .

The items will cost .

Alternatively: . You pay 80% of the original price, which is .

### 8.1.20 Going Down, Going Up

This activity involves working out increases and decreases together.

#### Activity: Going Down, Going Up

Let’s say you live and shop in Maryland, which in 2010 had a 6% sales tax. You have set your eyes on a new dresser for your room, which is on sale at 30% off. Have you ever asked yourself if it makes a difference if the discount is applied first, and then sales tax, or if it is done in the opposite order: Sales tax applied first, and then the discount?

Try a numerical example first to get a feel for the problem. For example, what happens if the price of the dresser was $100 (before the discount and sales tax have been applied)? ##### Answer If the item costs$100, reducing the price first gives .

The sales tax is 6% of $70, which is$4.20.

So, the final bill is .

Now, let's try it the other way.

Adding the sales tax first gives .

The discount is 30% of $106, which is . So, the total bill is . It does not seem to matter whether the discount or the sales tax is applied first. But can you be sure? Doing more numerical examples would confirm it. However, these could just be lucky picks. Let’s analyze what we can tell using these particular discount/sales tax rates. If the discount is 30%, the price after the discount will be of the original cost. You can find the discounted base price by multiplying the cost by 0.7. If the discount is applied first, then the sales tax, it’s 106% of 70% of the original price, and can be expressed as original price. If the sales tax is applied first, then the discount, it’s 70% of 106% of the original price, and can be expressed as original price. Since it does not matter in which order the multiplication operations are carried out, the answer will be the same using either method. This will work with any type of discount and sales tax. ### 8.1.21 Working Backwards In some situations, you may be given the total after the discount or tax has been applied and asked to find the original amount. For example, suppose an online store charges an extra 5% of the cost of the purchases for postage and packing. You have a gift voucher for$30. What is the maximum you can spend at the store, so that the total including the postage is less than $30? As you can see in the diagram above, 105% is equivalent to$30.

Therefore, 1% is equivalent to , and multiplying this quotient by 100 creates 100%, which results in (rounded down to the nearest cent).

The maximum amount that can be spent then is $28.57. You can check your answer by working out the 5% postage charge on$28.57 to see that the grand total will be $30. #### Activity: Working Backwards The price of a tennis racquet, including a 4% sales tax, is$45.24. What was the price before sales tax?

The $45.24 includes sales tax. What percentage does this total represent? Find the dollar amount that represents 1%, and use that figure to arrive at 100%. ##### Answer 104% of the cost is$45.24.

1% is and 100% is .

The combined amount in your checking and savings account is $35,300. Currently, you have credit card debt of$12,000 and still owe $11,200 on your school loans. Calculate your debt-to-equity ratio before and after the purchase. ##### Comment Have you identified the total amount of debt before the purchase? How about after purchasing the hot tub? ##### Answer Before: Your total debt (liability) is . Your equity (asset) is$35,300.

The debt-to-equity ratio without the purchase is .

After: [ Credit card debt is usually at a very high interest rate. You should seriously consider paying that off before making another large purchase! ]

The total debt would be . The equity would remain $35,300. So, the debt-to-equity ratio after purchasing the hot tub would be . Prior to the purchase, your debt to equity ratio is less than 1, which means you have more than you owe. However, if you decide to take out a loan to buy the hot tub, your debt to equity ratio becomes larger than 1, which means you owe more than you have. In other words, it’s probably not a wise decision to take out a loan to purchase this commodity! ### 8.2.3 Ratios in Recipes [ Feeling a little shaky with ratios written in colon notation? Check out this interactive website. ] So far, you have worked with ratios as fractions. Remember that ratios are also given in colon notation. For example, a recipe may call for two cups of raisins for every three cups of oatmeal. This can be written as the ratio 2:3. If you decided to make more or less of recipe, you need to preserve the ratio. In other words, there would need to be two parts raisins for every three parts oatmeal. Let’s look at an example. A recipe for shortbread requires 12 ounces of all purpose flour, 4 ounces of sugar and 8 ounces of butter. This means that the ratio of flour to sugar to butter is 12:4:8. Ratios can be canceled down like fractions. Dividing all parts of the ratio by 4, it can be expressed more simply as 3:1:2 or three parts flour, one part sugar and two parts butter. Suppose you wish to make some shortbread following this recipe using 12 ounces of butter, how much flour and sugar will you need? Start with the butter, 12 ounces is equivalent to two parts. So one part is the same as . Hence, three parts will be the same as . So, 18 ounces of flour and 6 ounces of sugar will be needed. ### 8.2.4 Ratios in Food and Drink #### Activity: Using Ratios in Food and Drink (a) A fruit drink has to be diluted by mixing 1 ounce of the concentrate with 5 ounces of water. How much water should be added to 4 ounces of concentrate? How much drink will this make? ##### Comment Have you considered drawing a sketch of a glass with ounces marked along its side? ##### Answer (a) The ratio of concentrate to water is 1:5, so five times as much water as concentrate is required. If 4 ounces of concentrate is used, of water is needed. If 4 ounces of concentrate and 20 ounces of water are used, the amount of drink will be . (b) The ratio of white flour to wholemeal flour in a bread recipe is 3:7. If 6 ounces of white flour is used, how much wholemeal flour is needed? Can you explain why this is called a “70% wholemeal loaf”? ##### Comment Because the ratio is 3:7, this means 6 ounces of white flour is equivalent to 3 parts. How much is 1 part of white flour? ##### Answer [ Alternatively, you could use the corresponding fraction and find an equivalent fraction where the numerator is 6 (because there are 6 ounces of white flour). ] (b) The resulting denominator would be your answer. If 3 parts are equivalent to 6 ounces, one part is equivalent to . 7 parts are equivalent to . Thus, 14 ounces of wholemeal flour are needed. There are 10 parts altogether, and 7 parts are wholemeal, so the fraction of the flour that is wholemeal is or 70%; hence the name. Notice that there are different ways of tackling these problems and sketches might help you to understand the situation better. ## 8.3 Extensions and Further Exploration In this section, we will try to extend our new knowledge through investigation of some of the mathematical content that was discussed throughout the unit. You might find some of these activities to be quite challenging. If you get stuck, feel free to discuss them with a friend. Don’t panic; just keep going. The goal of this section is to expose you to more types of math, and to help you realize just how often you already solve math problems in daily life. Remember, if you believe that you don’t have the time to spend further exploring these topics, this is a section that you could treat as optional. Now, get ready to learn the secrets of art and nature, and even bake some delicious cookies! ### 8.3.1 Percent Increase/Decrease Let’s take another look at percentages. In the following exploration, you will use the calculator to find a percent increase and a percent decrease. The calculator can be accessed on the left-hand side bar under Toolkit. #### Activity: Island Population The population of an island was 5,678 in the year 2010, with a prediction that it will rise by 3.1% over the next ten years (from 2010 to 2020). What will the population be in 2020? There are two ways you can look at this calculation. For the first method, you find the increase and add it to the original population, and for the second, you find the new population as a percentage of the original one. Method 1 involves finding 3.1% of the 2010 population of 5,678, then adding it to 5,678 to find the population in the year 2020. (a) Find 3.1% of 5,678, and then find the predicted population of the island in the year 2020. ##### Comment You learned how to find percentages of numbers earlier. ##### Answer (a) There are two methods for finding percentages of numbers, shown on the calculator screens here: 3.1% of 5,678 is 176. You need to round to a whole number because you can only have a whole number of people! The predicted population of the island in the year 2020 is . Method 2 is a little quicker. If the population rises by 3.1%, then the new population is , or 103.1% of the 2010 population. The population in 2020 is therefore 103.1% of 5,678. (b) Use this method to find the new population. ##### Answer (b) Using the calculator to find 103.1% of 5,678, you get: Or: In the first screenshot, the calculator cuts off the end of the answer, but the full answer is shown in the white screen below it. This method gives the same answer: The population of the island in 2020 is predicted to be 5,854. Now you try a percent calculation, a percent decrease this time. (c) The population on a neighboring island is falling. In the year 2010, it was 820, with a drop of 24% predicted over the next ten years (from 2010 to 2020). What will the population be in the year 2020? Try both methods shown above and make sure that they give the same answer. ##### Comment For method 1, find 24% of 820, and then subtract it from 820. For method 2, the new population is of the 2010 population. ##### Answer (c) Finding 24% of 820 on your calculator gives: So, the population drops by 197, and the predicted population in the year 2020 is then . For method 2, the new population is of the 2010 population. Using the calculator to find 76% of 820: The predicted population in the year 2020 is 623. #### To recap: To find the new amount after a percent increase, either: • find the increase and add it to the original amount, or • write the new amount as 100% plus the % increase, and find this percentage. To find the new amount after a percent decrease, either: • find the decrease and subtract it from the original amount, or • write the new amount as 100% minus the % decrease, and find this percentage. Now you can use your calculator for future percentage calculations. ### 8.3.2 Percent Decrease Whether it’s coupons you receive in the mail or a raise in your salary, you know that percentages are used to describe increases or decreases. For example, suppose last week a pair of sneakers cost$125, and this week the same pair costs $100. The price decreased by$25, which is referred to as the actual decrease.

To express the discount as a percentage, first write the actual decrease as a fraction of the original cost, then change the fraction into a percentage.

So the percent decrease is:

In other words, the price of the sneakers dropped by 20%. The process for calculating percent increases and decreases can be summarized as follows: .

### 8.3.3 Ups and Downs

#### Activity: Ups and Downs

(a) Gas prices rose from $3.25 to$3.51 a gallon since last week. What is the percent increase over the week?

##### Comment

What was the actual increase in price? What was the initial cost of gas?

(a) The actual increase is .

So, the percent increase in gas prices is .

(b) The 1981 Encyclopedia Britannica said that the Amazon Rain Forest occupied 2,700,000 square miles. Three decades later, Wolfram Alpha reported its total area to be 2.1 million square miles. Assuming each report was correct at the time of publication, find the percent decrease from 1981 to 2011.

##### Comment

What was the actual decrease in area? What was the original area occupied by the Amazon Rain Forest?

(b) The actual decrease is approximately .

Thus, the percent decrease in the amount of area covered by the Amazon Rain Forest is:

(c) You were just promoted at work and are getting a 3% raise. Congratulations! Your new salary, $56,780, reflects this increase. What was your salary before the raise? ##### Comment In this exercise, you already know the percent increase; we need to find the original or base salary. If$56,780 includes your raise, then it represents of your original salary. Should your original salary be less than or greater than your new salary?

(c) We know that $56,780 is 103% of the original salary. This directly translates to . To discover your original salary, we need to do the opposite of multiplication by 103%, which would be division by 103%. Thus, your original salary was: . (d) You just purchased a gas grill that was 35% off. The sale price was$312. What was the original cost of the grill?

### 8.3.4 Fact or Fiction?

Although percentages are used quite often in everyday life, they can be misleading and even a little tricky. In the next activity, you are asked to decide whether some statements are true or false, and to explain why.

#### Activity: Fact or Fiction?

For each of the following statements, say whether or not it is correct. Try working out some numerical examples to test your conjectures. Can you justify your results?

(a) If the price of a piece of jewelry is increased by 20% and then decreased by 20%, the item returns to its original price.

Pick any easy amount, like $100, to work with. Increase it by 20%, then take that new amount and decrease it by 20%. a. True b. False The correct answer is b. ##### Answer (a) Let’s try it using$100. If we apply a 20% increase, we will have 120% of the starting amount. 120% of $100 is . If we then apply a 20% decrease to this new amount, we will have 80% left. 80% of$120 is . So, the item does not return to its original price. The statement is false.

This happens for any price. To apply an increase of 20%, you multiply the price by 1.2, and then to apply a decrease of 20%, you multiply by 0.8. This is the same as multiplying by , which is 0.96. The new price will always be only 96% of the original price.

This also happens for any percentage. If, for instance, you apply an increase of 40% and then decrease it by 40%, the result is the same as multiplying by 1.4 and then by 0.6. With a 40% price increase and decrease, the end result will always be 84% of the original price.

(b) If the price of a pair of jeans is decreased by 20%, and two pairs are purchased, then the total paid is a 40% decrease from the original price.

##### Comment

Select an easy number to work with and see what happens!

a.

True

b.

False

(b) Increasing a number by a certain percent and then decreasing it by the same percent will not result in the original number.

Suppose the pair of jeans costs $100. If we apply a 20% reduction, the new price will only be 80% of the original cost. 80% of$100 is . So, if you buy two pairs at $80 each, the total is$160 instead of $200, a reduction of$40.

The overall percent decrease is therefore . Thus, the statement is not correct.

To help convince you that this is the case, a diagram might help. You can see that the discount on the two items together is still only 20% of the total price.

### 8.3.5 Percentage Points

#### Percentage Points

When comparing percentages, the difference between the percentages is described in terms of “percentage points.” Subtracting percentages gives percentage points.

[ A mortgage point is 1% of the amount you borrow for the purchase of your house. It is an additional fee that the bank may collect in return for offering a reduction on the annual percentage rate (APR) of the mortgage—in other words, paying this fee entitles you to a cheaper loan term. ]

Note that this is not the same as the percent increase or decrease. This distinction is something to be aware of in media reports, particularly when interest rates or buying mortgage rate points are discussed.

For example: If 72% of students starting at a community college were placed in a noncredit math course one year, but only 63% the next, the college had a decrease of 9 percentage points in students not being ready for credit math courses.

The next activity illustrates the difference between percentage points and percent change.

#### Activity: Student Loans

You took out a private student loan for college fees. You see a headline reading, “Interest rates jump 2%.” Should you be worried?

##### Comment

Most people read this headline to mean that the interest rate on the loan increases by 2 percentage points. What would that mean for you?

Is there another way to think about the headline that isn’t as sensational?

The headline is ambiguous. It could mean “Interest rates increase by 2 percentage points”—for example, from 10% to 12%. This is a big deal. Your interest rate just went up 20%: .

On the other hand, the headline could mean “Interest rates increase by 2% based on your previous interest rate.” In our example, that would mean rates go up from 10% to of

Well, OK, not great news, but hopefully manageable and much better than our first interpretation.

When people mean percentage points, they should say so! Unfortunately, many news articles do not. It’s up to you to find out from the article itself what it is about.

In newspapers and magazines, you will often run into articles that contain percentages. The exercise below illustrates the importance of interpreting this information correctly.

#### Activity: Percentage Points

A newspaper headline reads “Exam pass rate increases from 50% last year to 75% this year.”

Does this mean that the pass rate has increased by 25%? Explain why or why not.

##### Comment

Did you convert the percentages to their decimal values before using the percent increase formula?

Writing the pass rates in decimal form as 0.5 and 0.75, the percent increase is:

So, the pass rate has increased by 50%, not 25%. However, it would be correct to say that there was a 25 percentage point increase in the exam pass rate.

The bottom line is to be sure that you pay attention when you run across percentages. A statistic might not mean what you initially think it does. Keep your eyes open and your brain turned on!

### 8.3.6 The Golden Ratio

Have you ever heard of the golden ratio?

[ To learn more about the golden ratio and play with the proportion, visit this site. ] One way to express the golden ratio is , which is approximately 1.61803398. This value describes the ratio found in many contexts, such as number patterns, geometry, and nature.

Some people claim that measurements in paintings, aesthetically pleasing architecture, and even the human body reflect the golden ratio! Check out this short video to see some examples and claims of where the golden ratio seems to appear:

Interactive feature not available in single page view (see it in standard view).

Some artists and architects have deliberately used the golden ratio in their work, but often the ratio of measurements is only approximately equal to the golden ratio. Even some of the claims made in the video you have just watched have been disputed: see, for instance, mathematician Keith Devlin’s article, “The Myth That Will Not Go Away.”

When you see a claim that two measurements are in the golden ratio, be critical. Check the measurements and work out their ratio. You may get an answer that is approximately the value of the golden ratio, but this might have occurred by chance.

There have been many claims that the golden ratio occurs in the proportions of the human body. You can test this claim in the next activity!

#### Activity: The Golden Ratio and the Human Body: A Match Made in Heaven?

For this activity, you will need a measuring tape and a buddy.

(a) Stand with one arm out to your side. Have your buddy measure the distance from your middle finger tip to your elbow. Record this value. Using the same unit of measurement, let your friend measure from your wrist to your elbow. Record this value, too. Now, calculate the ratio of the first measurement to the second.

(b) Repeat part (a) with measurements for your friend.

(a) and (b)

The ratios are probably somewhere between 1.5 and 1.7. However, there are variations in the proportions of the human body for different people, so you may have values different to these. (The value should be greater than 1 though, so if it isn’t, check your calculation!)

(c) It has been claimed that the ratio you calculated in part (a) and part (b) will reflect the golden ratio. Do you agree?

(c) It is unlikely that the ratios are exactly equal to the golden ratio, 1.618. The ratio depends on how precisely you made the measurements and where from. For example, which part of the wrist did you use? If you repeated the measurements, would you get the same results? Sometimes a small change in the measurement can appear to give a ratio closer to the golden ratio. Although each of your ratios may be approximately equal to the golden ratio, this may just be a coincidence!

### 8.3.7 Sweet or Savory?

You have just finished working through a unit that involves some fairly sophisticated concepts. Kudos to you! Now it’s time to reward yourself!

#### Sweet or Savory?

Do you have a sweet tooth for chocolate, or are you craving melted cheese? Below are two recipes for you to choose from.

Interactive feature not available in single page view (see it in standard view).

Pizza

Interactive feature not available in single page view (see it in standard view).

## 8.4 Self-Check

### 8.4.1 Self-Check on Percentages

In this unit, you might have struggled with certain activities, and that’s okay! We all get stumped from time to time. Remember that there are plenty of resources that can provide guidance and advice, such as family, friends, teachers, old textbooks, and the Internet.

It’s important for you to reflect on the material you have studied and consider the development of your skills. Your confidence will continue to increase the more you practice. You can do this by working through the exercises for each section. Here are some activities to help you perform a self-check.

#### Exercise 1: Fractions, Decimals, and Percentages

Copy the table below into your notebook. Then, fill in the columns to show which fractions, decimals, and percentages are equivalent to each other. The first column has been filled in for you. You have worked out some of these calculations already earlier in this unit.

Equivalent fractions, decimals and percentages
PercentageDecimalFraction
5%0.05
10%
0.2
0.5
75%
1
##### Comment

To change a fraction or decimal to a percentage, multiply it by 100%.

For example, and .

To change a percentage to a fraction or decimal, interpret the % symbol to mean “divided by 100”.

For example, and .

To change a fraction to a decimal, divide the numerator by the denominator.

For example: .

Equivalent fractions, decimals, and percentages
PercentageDecimalFraction
5%0.05
10%0.1
20%0.2
25%0.25
30%0.3
50%0.5
64%0.64
75%0.75
100%11

Note: Converting a decimal into percent means moving the decimal point two places to the right and including the percent symbol.

Note: In your conversion from a fraction to a percent you can also go through the decimal, i.e. work out the quotient and then move the decimal point two places to the right and include the percent symbol.

#### Exercise 2b: Sales Figures

In January, a company sells $8,500 worth of goods. However, the sales fall by 8% in February. How much is sold in February? ##### Answer 8% of$8500 is .

The total sales in February are .

Alternatively, 92% of $8500 is . #### Exercise 2c: Sales Tax An item costs$35.49, plus sales tax. If the sales tax rate is 6%, what is the total price?

6% of $35.49 is in sales tax. Together it is . The item costs$37.62 after sales tax.

Alternatively, .

#### Exercise 3: A Clerical Error

A mail order company offers a 15% reduction on its prices to new customers. Unfortunately, a clerical error has been made and some existing customers have also been given the discount. One existing customer has been charged $144.50, including the discount. What should the customer have been charged? How can you check your answer? ##### Answer The reduction is 15%, so eligible customers are charged 85% of the original cost. 85% of the cost is$144.50.

1% of the cost is and 100% of the cost is .

The existing customer should have been charged $170. You can check this answer by calculating the reduced price. 15% of$170 is .

So, the reduced price is .

#### Exercise 4: Large Group at a Restaurant

In 2010, you went on a trip to visit some friends in Montana, which charged no sales tax at the time. You and six friends went out to dinner together—a party of seven in all. The restaurant’s policy stated that groups of six people or more will have 18% gratuity (tip) added to their bill. If your food and drinks totaled $176.23, how much was the bill? If you had split the bill evenly seven ways, how much was your share? ##### Answer 18% of 176.23 is and the total bill was . Divide by seven and is what you and each other person in the party owed. #### Exercise 5: “Of” versus “Off” (a) What percentage of the original price do you pay if an ad promises “20% off everything?” ##### Answer (a) You pay 80% of the original price. (b) What percentage of the original price do you pay if “the sale price is 40% of the original price?” ##### Answer (b) You pay 40% of the original price. #### Exercise 6: Discount Find the sale price of a$340 television that is on sale for 25% off.

25% of $340 is . With a discount of$85, the sales price of the television is .

#### Exercise 7: Time for That New Television

The price of a 55-inch flat-screen television is $1,897.22, which includes 8.25% California sales tax. How much did the television cost before sales tax? ##### Answer$1,897.22 is 108.25% of the original price. Dividing the after sales tax amount by 108.25% gives you 1% of the original price: . Multiply this result by 100 to create 100%, and thus the original price, which is (rounded to the nearest cent).

The pretax price of the television was $1,752.63. You can check your answer by finding out what a television of$1,752.63 would cost after 8.25% of sales tax is applied.

#### Exercise 8: Spending

The consumer expenditure survey (Graph: Where the Money Goes) shows how much the average household spent on different categories in 2009. The percentages displayed in the picture were calculated based on the total money spent (average household expenditure).

If based on a different number, these percentages will be different. According to the numbers in this picture, what percent of the average household’s income before taxes ...

(a) ... is spent on food away from home?

(a) Food away from home: .

(b) ... is spent on housing?

(b) Housing: .

(c) ... is not spent?

(c) Money not spent is taxes, savings, etc. This is the income before taxes minus average annual expenditure, . As a percent of the income before taxes, this is .

### 8.4.2 Self-Check on Ratios

#### Exercise 9: Chocolate Chip Cookies

A recipe for chocolate chip cookies requires 48 tablespoons of flour, 16 tablespoons of sugar, and 32 tablespoons of chocolate chips.

(a) What is the ratio of flour to sugar to chocolate chips?

(a) 48:16:32 which reduces to 3:1:2 (through division by 16).

(b) If you only have 36 tablespoons of flour, how many tablespoons of sugar and chocolate chips do you need to maintain the recipe’s taste and consistency?

(b) 3 parts of flour is equivalent to 36 tablespoons. So, 1 part is equivalent to 12 tablespoons. Thus, you will need tablespoons of sugar and tablespoons of chocolate chips.

(c) Suppose you have 40 tablespoons of chocolate chips, and plenty of everything else. Determine the amount of flour and sugar you will need, if you want the maximum number of cookies you can make with your 40 tablespoons of chocolate chips.

In this case, 2 parts of chocolate chips will be equivalent to 40 tablespoons. So, 1 part is equivalent to 20 tablespoons. Thus, you will need tablespoons of flour and tablespoons of sugar.

## 8.5 Quiz Time

### Practice Quiz

Now that you have taken the time to work through these sections, consider giving this short quiz a try! You may find that it will help you to monitor your progress, particularly if you took the quiz at the start of the unit as well.

The quiz does not check all the topics in the unit, but it should give you some idea of the areas you may need to spend more time on. Remember, it doesn’t matter if you get some, or even all of the questions wrong—it just indicates how much time you may need to come back and review this unit!

## 8.6 Study Checklist

### Study Checklist

Take stock of the material you covered in this unit. Below is a list that highlights the main ideas you’ve studied. This was a challenging unit, but, if you keep your eyes open, you will be reminded of these concepts.

You should feel confident to:

• Write numbers as percentages, fractions, and decimals.
• Perform calculations with percentages.
• Understand how percentages and ratios are used in everyday life.
• Convert between fractions, decimals, percentages, and ratios.
• Apply percentages and ratios to given scenarios.
• Revisit some of the strategies that help you to solve problems.
• Continue to appreciate how to write good mathematics.

You’ve come a long way. Well done!

### Outlook on Unit 9

Let’s move onto Unit 9, where there’s some exciting material on patterns and formulas, as well as news about an unsolved math problem with a \$1,000,000 prize attached! But mathematicians have been trying to solve this one since 1742, so don’t get your hopes up too high …