Skip to main content
Printable page generated Saturday, 23 Oct 2021, 20:38
Use 'Print preview' to check the number of pages and printer settings.
Print functionality varies between browsers.
Printable page generated Saturday, 23 Oct 2021, 20:38

Unit 3: Everyday Math

3 Introduction

In this unit you will investigate the four basic operations—addition, subtraction, multiplication, and division—including the math language associated with them. You will also get the chance to find out about different strategies to help you deal with mental math problems.

Practice Quiz

To check your understanding of these operations before you start, give the Unit 3 pre quiz a try, then use the feedback to help you plan your study.

The quiz checks most of the topics in the unit, and should give you a good 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!

Click here for the pre quiz.

So far, we’ve studied how to write, estimate, and round numbers. Now we will begin to perform calculations with them. First, let’s look at addition and subtraction of numbers.

3.0.1 What to Expect in this Unit

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

  • Addition.
  • Subtraction.
  • Multiplication.
  • Division.
  • Mental math strategies using these operations.

3.1 Addition and Subtraction

One of the most important advantages of the decimal number system is that calculations can be carried out easily as everything is based on the number ten. This section briefly considers the fundamental operations of addition (+) and subtraction (–) and shows how they can be used in tackling problems. Although you will be using your calculator for a lot of calculations, there will be occasions when it is useful either to work out an answer to a calculation on paper or to carry out a quick calculation in your head. If you are confident in your addition and subtraction skills, then skip this section and go on to the next.

3.1.1 Addition

If you are working out a budget, checking a bill, claiming a benefit or work expenses, determining the distance between two places by car, or many other everyday calculations, you will probably need to add some numbers together. Consider the following questions.

  • How much is the bill altogether?
  • How much will your vacation cost now that the total price of your airfare has increased by $50 due to fuel surcharges?
  • Five more people want to come on the trip: What is the total number of people booked on the bus now?
  • What is the cost plus sales tax?

All the questions above involve addition to work out the answer. The key or “trigger” words that indicate addition have been bolded. The process of adding numbers together may also be referred to more formally as finding the sum of a set of numbers.

When you are adding two numbers together, it does not matter in which order you add them together, so two plus three gives the same answer as three plus two. In mathematical terms, we say that addition is commutative. Not every operation will possess this property (for example five minus three not equals three minus five).

Keep writing down important math language in your notebook with your own definitions—this will help you to remember them!

Let’s take a closer look at how to add numbers.

Adding Numbers

To add numbers in our decimal system, write the numbers underneath each other so that the decimal points and the corresponding columns (or placeholders) line up. Then add the numbers in each column, starting from the right.

[ Remember that when you are adding, if the sum is larger than 10, you will need to regroup (or carry) into the next place holder. ] For example, let’s add 26 full stop two plus four zero eight full stop 75 plus 0.0 seven:

Pencast symbol If it’s been a while since you’ve worked with addition, especially with decimals, please check out the Adding Numbers pencast. This  explains how to regroup or carry (Click on “View document”).

View document

Addition in the Real World

Activity symbol Activity: Checking a Bill

Suppose you buy three items costing $24.99, $16.99, and $37.25 from a mail order catalogue, and the postage is $3.50. Round these prices to the nearest dollar and work out an estimate for the total bill in your head, or in your math notebook if you prefer. Then check your estimation by writing out the problem without rounding and performing the calculation again to give the accurate bill.

Hint symbol
Discussion

Remember to follow the pattern above. Find estimates for $24.99, $16.99, $37.25, and $3.50, then add them up in your head or on paper. Next, write down the exact numbers in a vertical stack, lining up the decimals and place values.

Solution symbol
Answer

You may have your own method of working things out in your head that works fine for you. If so, there is no need to change how you do this, but you might find the following method useful to work through. The four rounded prices are $25, $17, $37, and $4. To find an estimate for the total cost, add the four rounded prices together.

This is easier to do if you split each number apart from the first one, into tens and units, so 17 can be split into 10 plus seven and 37 into 30 plus seven. This makes the sum much easier to work out in your head and the sum then becomes sum with, 6 , summands 25 plus 10 plus seven plus 30 plus seven plus four.

Then working from the left and adding each number in turn, you can say, “25 and another 10 makes 35; another 7 gives 42; another 30 gives 72; 7 more makes 79, and another 4 gives 83.” The answer is 83, so the total bill will be approximately $83.

To check your estimate on paper, you will need to line the unrounded numbers up and then add them.

You can see that our estimate of $83 was very close to the exact amount $82.73. Now you can be confident in your answer. Good work!

Mental Math

Activity symbol Activity: All in Your Head

Try working out the following problems in your head using the strategies from the previous activity.

(a) six plus 17

Hint symbol
Discussion

Consider breaking down numbers to make easy addition problems.

For example,equation sequence seven plus 24 equals sum with, 3 , summands one plus six plus 24 equals one plus 30 equals 31. Splitting the numbers up like this means that you can work in tens, which makes them much easier to add up in your head.

Solution symbol
Answer

(a) Here are two different approaches you might try:

multiline equation row 1 six equation left hand side equals right hand side three plus three row 2 17 equation left hand side equals right hand side 10 plus seven row 3 13 plus 10 equals 23

Or, equation sequence six plus 17 equals 17 plus six equals sum with, 3 , summands 17 plus three plus three equals 20 plus three equals 23.

(b) 25 plus 13

Solution symbol
Answer

(b) equation sequence 25 plus 13 equals sum with, 3 , summands 25 plus 10 plus three equals 35 plus three equals 38

(c) 29 plus 19

Solution symbol
Answer

(c) Here, we could say that 29 is one less than 30, and 19 is one less than 20, so 29 plus 19 is two less than 30 + 20, or 50. So, the answer is 48.

Alternatively, you could write equation sequence 29 plus 19 equals 30 minus one plus 20 minus one equals 50 minus two equals 48.

(d) sum with, 5 , summands 12 plus five plus eight plus three plus two

Solution symbol
Answer

(d) equation sequence sum with, 5 , summands 12 plus five plus eight plus three plus two equals open 12 plus eight close plus open sum with, 3 , summands five plus three plus two close equals 20 plus 10 equals 30.

Here instead of spitting the numbers up to help us work in tens we have grouped them together to give us tens.

Pencast symbol Want a little more practice with mental math? Check out this pencast on mental math (Click on “View document”).

View document

You can practice these mental techniques in lots of different ways, but it is a good idea to start with easier problems and/or situations before you move forward. Get comfortable with the process and never be afraid to make mistakes. That’s how you learn! You might like to practice with your friends and family and see how you get on together.

3.1.2 Subtraction

Consider the following questions:

  • What’s the difference in distance between taking the Eighth Street Bridge versus MLK Boulevard?
  • How much more money do we need to save?
  • If we take away 45 of the plants for the front garden, how many will be left for the back garden?
  • If all vacation prices have been decreased (or reduced ) by $20, how much is this one?

All these questions involve the process of subtraction to find the answer—a process that you often meet when dealing with money. You can see that the “trigger” words for subtraction are again in bold.

One way you can think of subtraction is as the process that undoes addition. For example, instead of saying “$10 minus $7.85 leaves what?” you could say, “What would I have to add to $7.85 to get to $10?”

Adding on 5 cents gives $7.90, another 10 cents gives $8, and another $2 will give you a total of $10. So, the total amount to add on is equation left hand side sum with, 3 , summands dollar times 0.05 plus dollar times 0.10 plus dollar times 2.00 equals right hand side dollar times 2.15. This is the same answer as the one obtained by subtraction: equation left hand side dollar times 10 minus dollar times 7.85 equals right hand side dollar times 2.15.

In other words, you can always check or work out a subtraction problem by using addition. For example, if you calculate 1874 – 1476 = 398, then a quick check is to make sure 1476 plus 398 equals 1874, which it does.

Working out calculations like this on an everyday basis may help you to feel more confident in working with numbers. You can always check your answer on a calculator, if you like.

Subtraction in the Real World

Activity symbol Activity: Making Change

Suppose you have $25.90 in cash (two $10 bills, five $1 bills, and nine dimes) and you owe a friend $16.87. How will you break the $25.90 in order to pay off your debt?

Hint symbol
Discussion

Start with the $16.87. You can give your friend a $10 bill. What will you still owe her?

Solution symbol
Answer

After giving your friend $10, you will still owe her equation left hand side dollar times 16.87 minus dollar times 10.00 equals right hand side dollar times 6.87. You still have five $1 bills, so you fork those over, too! Now you only owe her equation left hand side dollar times 6.87 minus dollar times 5.00 equals right hand side dollar times 1.87.

Since your friend won’t budge, you need to make change for your remaining $10 bill. Let’s say you break it into a $5 bill and five $1 bills. Then you hand over $1 to your friend.

At this point, your debt is now equation left hand side dollar times 1.87 minus dollar times 1.00 equals right hand side dollar times 0.87. After giving your friend eight dimes, the amount you still owe is equation left hand side dollar times 0.87 minus dollar times 0.80 equals right hand side dollar times 0.07.

Taking your last dime and trading it for a nickel and five pennies, you hand over the last 7 cents you owe. Now you’ve erased your debt!

Subtracting One Number from Another

[ Remember that if you don’t have enough in one column to perform the subtraction, you will redistribute (or borrow) from the place holder to the left. ] Let’s determine how much money we’d have left over from the previous example.

Write out the calculation so that the decimal points and the corresponding columns line up, making sure the number you are subtracting is on the bottom. Then, subtract the bottom number from the top number in each column, starting at the right and changing the amounts in each column as needed.

Pencast symbol If you would like a more detailed explanation of how to perform subtraction using this method, take a look at the subtraction pencast (Click on “View document”)

View document

You may already be familiar with other ways of subtracting numbers. This is fine; just make sure that you can use either this method or your own confidently. Remember, you can check your answer ($9.03) by adding it to the number you subtracted ($16.87)—you should end up with the number you started with ($25.90).

Activity symbolActivity: Subtraction

Carry out the follow subtractions on paper.

(a) 526 – 327

Solution symbol
Answer

(a) 526 – 327 = 199

(b)1029 – 822

Solution symbol
Answer

(b) 1029 – 822 = 207

(c) 776 – 389

Solution symbol
Answer

(c) 776 – 389 = 387

(d) 429 – 338

Solution symbol
Answer

(d) 429 – 338 = 91

How did you get on with that activity?  If you think that you need to practice these skills further try applying them to real-life situations, making up some examples of your own or asking friends and family for some.  You can check the answers on your calculator.

3.2 Time, Addition, and Subtraction

Activity symbol Activity: Train Journeys

To get to a very important meeting you can travel by train, leaving at 9:35 a.m. and arriving at 11:10 a.m. The ticket costs $48.30, but you have a refund voucher for $15.75 following the cancellation of a train on an earlier journey. You know from experience that the conductor won’t let passengers leave the train for an additional ten minutes after arriving at the station.

How long will the journey take, and how much money will you have to pay for the ticket if you use the entire refund voucher?

Hint symbol

Discussion

Start by taking out all the information from the question—this should help to make the problem clearer.

To determine the length of time, you could figure out how many minutes you will have been on the train by 10:00 a.m., and then add up the time until you reach 11:10 a.m. Don’t forget the extra ten minutes the conductor requires.

To determine your total cost, write down the ticket cost and subtract your refund. Use the vertical format, where your digits are lined up by place value and the decimal points are aligned.

Solution symbol

Answer

Working with times can be tricky, because you have to remember that there are 60 minutes in an hour, so you are not working with the simple decimal system. Imagine a clock and say, “From 9:35 to 10:00 is 25 minutes; 10:00 to 11:00 is an hour, and 11:00 to 11:10 is an extra ten minutes. So the journey time is 1 hour and 35 minutes, assuming the train runs on time. Then the conductor requires waiting an extra ten minutes, giving a grand total of 1 hour and 45 minutes.”

Now you have one more step:

To find out how much you will have to pay for the ticket, you need to subtract $15.75 from $48.30. Think it out this way:

“Doing a formal subtraction in my head isn’t easy, so I’ll work up from $15.75. Adding on 25 cents gives me $16.00, then I need $32 to get to $48.00 and another 30 cents to reach $48.30. So, the total I have to pay is $32 plus 25 cents plus 30 cents, or $32.55.”

Now you have learnt about addition and subtraction, and some ways of tackling problems involving these, let’s investigate the two other operations: Multiplication and division.

Remember to continue to think about how your studying is going and to use your math notebook to jot down your thoughts.

3.3 Multiplication and Division

In the previous section, we noted that the processes of addition and subtraction can be described in several different ways in questions and problems. Let’s begin by doing the same with multiplication and division.

Here are some examples of different words that are used for multiplication:

  • 5 times 6.
  • 3 lots of 20.
  • The product of 6 and 7.
  • You might see on an automated teller machine: “Please enter a multiple of $20.”

All the words in bold are again “trigger” words telling us we need to multiply.

Can you suggest some practical situations when this operation is used? Here are some we came up with:

  • Cooking, such as doubling the ingredients needed for a recipe or cutting them in half.
  • To work out the cost of multiple items—eight books at $4.99 each, say.
  • Calculating your daily income based on your hourly pay rate.

Now let’s take a look at division.  Here are some examples of different words and phrases that are used for division:

  • How many times does 8 go into 72?
  • How can we share 72 items equally among eight people?
  • How many lots of  five are there in 20?

Can you suggest some practical situations when this operation is used?  Here are some we came up with:

  • Dividing a bill equally among a given number of people.
  • Sharing slices of pizza equally among friends.
  • Working out how many measures of liquid there are in a container.

Ask your friends and family how they use multiplication or division.

3.3.1 Multiplication and Division Notation

The multiplication symbol was first used by an English mathematician William Oughtred (1574–1660) in 1631. Another symbol for multiplication is asterisk operator, although this is generally only used when writing math for computer programmes.

[ The division symbol (division) was first used by Johann Rahn (1622−676) in a book published in German in 1659. ] For division, a range of notations are used: 72 division eight, or 72 divided by eight, or 72/8 or eight times 72 all mean “72 divided by 8.”

3.3.2 Checking Your Math

You will often perform multiplication and division on your calculator. However, you should be able to check your answers by hand just in case you don’t have access to a calculator, or in case you incorrectly key in a number on your calculator. Remember that a calculator is only as good as its user is at inputting information!

In order to check that your calculations are reasonable, it is useful to be familiar with multiplication and division by the digits 2 through 10 and 100. Knowing the results of multiplying by the numbers from 2 to 10 can help you to work out calculations fairly quickly and confidently. Try this times tables grid game to see how you get on and to brush up on your multiplication tables.

Having the multiplication tables at your fingertips really helps and the only way for most of us to do this is with practice. You can use spare moments in your day to do this as well as asking friends or family to join you in your task!

Before we will look at some methods that might be useful when multiplying or dividing, let’s make sure that we understand the math language associated with these.

3.3.3 Multiplication and Division Math Terminology

Let’s review the pieces of a multiplication or division problem to be sure we are using the correct vocabulary. All the words in bold are important math vocabulary, so add them to your math notebook with your definitions as you come across them.

Suppose we have six times normal x times four equals 24. The 6 and 4 are referred to as factors, and 24 is called the product. The numbers being multiplied together are the factors, and the answer to a multiplication problem is the product. So, factor multiplication factor equals product.

Now, say we have 56 division seven equals eight. The name for the 56 in this example is dividend, 7 is the divisor, and 8 is the quotient. The number being divided, or broken down, is the dividend. The number we divide by is the divisor, and the answer to a division problem is the quotient. So, dividend division divisor equals quotient.

The remainder in a division problem is the number “left over” that the divisor can no longer divide into. We say a divisor evenly divides a dividend if the remainder is zero. However, if there is a remainder, then the common notation is to write the quotient, then R, followed by the value. For example, equation left hand side eight division five equals right hand side one times cap r times three, because 5 goes into 8 one time, which gives us 3 left over.

We can also relate division to multiplication. We say 56 division seven equals eight because eight multiplication seven equals 56. This shows that division undoes or is the opposite of multiplication.  We can use this rule to show that division is the only basic arithmetic operation that has a restriction. Consider six division zero. Let’s say six division zero equals a. This is equivalent to saying that a multiplication zero equals six, but 0 times any number results in 0, so a multiplication zero equals six is impossible. Thus, division by zero is not defined; it does not exist.

However, zero can be divided by any non-zero number. For example, zero division eight equals zero, because eight multiplication zero equals zero full stop Zero can be a dividend or a quotient—just not a divisor.

Activity symbol Activity: Find the Quotient

(a) What is the quotient of 30 division three?

Hint symbol

Discussion

Use multiplication. For example, three postfix multiplication question mark equals 30.

Solution symbol

a. 

A. 27


b. 

B. 10


c. 

C. 6


d. 

D. 9


The correct answer is b.

Answer

(a) Because three multiplication 10 equals 30, the correct answer is 10, which is letter B.

(b) What is the quotient of 12 division four?

Solution symbol

a. 

A. 8


b. 

B. 10


c. 

C. 3


d. 

D. 2


The correct answer is c.

Answer

(b) Because four multiplication three equals 12, the correct answer is letter C.

3.3.4 Relationships Among Operations

Multiplication can be thought of as repeated addition. For example, we know that two multiplication three equals six. You probably memorized this multiplication fact as a child, but why is this true? Well, we can think of multiplication as a direction to add the first number to itself the second number of times. So, two multiplication three is telling us to add 2 to itself 3 times: sum with, 3 , summands two plus two plus two equals six.

Similarly, division can be thought of as repeated subtraction. For example, we know that 20 division four equals five. The answer is 5 because 5 is the number of times you subtract 4 from 20 to arrive at zero:

Videoclip symbolLet’s review how to carry out multiplication and division of larger numbers as well as hone our mental math skills by watching this seven-minute video. NOTE: At 3:12 the professor incorrectly calls the dividend the quotient. Even teachers make mistakes!

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

3.3.5 Performing Multiplication and Division

Activity symbol Activity: Multiplication and Division

In your math notebook, perform the following operations by hand.

(a) 348 multiplication 37

Hint symbol
Discussion

For multiplication, begin by writing the problem vertically and lining up the digits in the units place. For division, set the problem up using multiline equation line 1 for the division symbol.

Solution symbol
Answer

(a) multiline equation line 1 348 line 2 separator multiplication 37 macron line 3 2436 line 4 plus plus 10440 macron line 5 12876

Thus, the answer is 12,876. (This seems reasonable since 300 multiplication 40 equals 12000.)

(b) 560 times normal x times 23

Solution symbol
Answer

(b) multiline equation line 1 560 line 2 separator multiplication 23 macron line 3 1680 line 4 plus plus 11200 macron line 5 12880

Thus, the answer is 12,880. (This seems reasonable since600 times normal x times 20 equals 12000.)

(c) 4968 division 24

Solution symbol
Answer

(c) multiline equation line 1 24 times multiline equation line 1 207 line 2 4968 line 2 zero width space zero width space zero width space zero width space zero width space zero width space zero width space separator minus minus 48 low line line 3 16 line 4 separator minus zero low line line 5 168 line 6 minus minus 168 low line line 7 zero

So the solution is 207. (This seems reasonable because 5000 division 25 equals 200.)

(d) 4035 division 15

Solution symbol
Answer

(d) multiline equation line 1 269 line 2 15 times 4035 line 3 minus 30 macron line 4 prefix plus of 103 line 5 minus 90 macron line 6 135 macron line 7 minus 135 macron line 8 zero

So the solution is 269.

Videoclip symbolIf you feel like you might need some extra practice with long division, check out this video. You will be taken through two examples step by step.

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

3.3.6 Multiplication and Division Strategies

Let’s try to figure out some shortcuts for when you are having trouble remembering the multiplication tables! Here are some strategies that you might find useful. You must remember the rules behind each one and try and practice those techniques that work best for you.

Multiplication:

  • To multiply by 5, you can multiply by 10, and then divide the product by 2.
  • To multiply by 4, you can double the other factor, then double it again.
  • To multiply by 8, you can double the other factor three times.

Division:

  • To divide by 4, you can divide by 2, and then divide by 2 again.
  • To divide by 5, you can divide by 10, then multiply by 2.
  • You might find that your friends or family have some other ideas they find helpful – why not ask them?

3.3.7 Commutative Property

Remember we said in Unit 2 that when you add two numbers together, the order does not matter—the same as saying that addition is commutative; so two plus four is the same as four plus two. But, what about multiplication and division? Is three multiplication two the same as two multiplication three? Is four division two the same as two division four? What do you think and how can you convince yourself that your answers are right? When you have given this some thought see below!

When you multiply two numbers together, the order does not matter. So, three multiplication two is the same as two multiplication three. Look at the diagram above, which shows on the left three rows of two dots (3 x 2). Turn this around so that it shows two rows of three dots (2 x 3). The number of dots in both arrangements is the same, 6, and hence you can see that equation left hand side three multiplication two equals right hand side two multiplication three.

You can also check it by using the idea that multiplication is repeated addition. Remember that equation sequence three multiplication two equals three plus three equals six, and equation sequence two multiplication three equals sum with, 3 , summands two plus two plus two equals six. This means you can carry out the calculation in whichever order you find easier. Multiplication, like addition, is therefore commutative.

However, the order you carry out division does matter: four division two is not the same as two division four. For example, if we divide $4 between two people, each person gets $2. If instead we need to divide $2 among 4 people, each person only gets $0.50. So division is not commutative.

3.3.8 More Division Strategies

Admittedly, division is usually more difficult than multiplication, but you can help yourself by having a good knowledge of the multiplication tables. In addition to this there are some other approaches we might use when dividing to help.

1. Using Groups of the Divisor

One way of looking at division problems is to see the problem as a question about the number of groups. For example, if the problem is 159 division 13, you may think, “How many groups of 13 are in 159?”

This approach uses multiplication to create groups of the divisor, keeping track of what part of the dividend remains. You may think about ten groups of 13, or 10 multiplication 13 equals 130. Next, you would subtract 130 from 159 mentally, leaving 29. Recognize that two more groups of 13 can “fit” into what remains, accounting for 26 more with only three left over. So 13 goes into 159 a total of 10 plus two equals 12 times, with three left over. To check that, multiply 12 multiplication 13 equals 156, and 156 plus three equals 159. Bravo!

2. Breaking the Dividend into Parts

In this strategy, think about the dividend first, and how it can be broken up into numbers that are easier to divide by the divisor. For example, for the problem 159 division 13, you could think: “How can I break 159 apart to make this an easier problem to solve?”

You might think of this as:

multiline equation line 1 equation left hand side 159 division 13 equals right hand side open 130 division 13 close plus open 29 division 13 close line 2 equation left hand side equals right hand side open 10 close plus open 29 division 13 close line 3 equation left hand side equals right hand side sum with, 3 , summands 10 plus two plus open three division 13 close line 4 equation left hand side equals right hand side 12 r times e times m times a times i times n times d times e times r 13 line 5 equals 12 cap r times three

Of course, there are many other strategies. Let’s look at two more that you might find useful in different situations.

3.3.9 Equivalent Problems (Division Strategies)

3. Making an Equivalent Problem

This is similar to the way that fractions are simplified. For example, in the math problem equation left hand side 1400 division 35 equals right hand side 1400 divided by 35, both the top and bottom of the fraction can be divided by 35. You will have the opportunity to study this in Unit 6.

A division problem can be changed into an equivalent problem by dividing both numbers by the same number, thus maintaining the quotient between them.

Consider the division problem 1400 division 35.

Thus, 1400 division 35 equals 40.

3.3.10 Using a Related Problem

4. Solving an Easier, Related Problem and then Compensating

You can solve a problem where you already know the answer and adjust. For example, when solving 159 division 13, you may know off the top of your head that 13 multiplication 13 equals 169, and then you might reason that subtracting 13 from 169 is 156, the closest multiple of 13 to 169, without going over 159. In other words, 12 multiplication 13 equals 156, which is three less than 159, so 159 division 13 equals 12 cap r times three.

Activity symbol Activity: Paper Supplies

A college bookstore buys pads of legal paper in bulk to sell to students in the law program at a cheap rate.

Each pack of paper contains 20 pads. If the store wants 1500 pads for the term, how many packs should be ordered?

Hint symbol
Discussion

We need to find how many groups of 20 are in 1500, so the calculation is:

20 times 1500

If you imagine dividing some quantity of objects into 20 piles, one way to do it would be to divide it into ten piles instead, and then divide each of those piles in half. So, dividing a number by 20 is the same as dividing by ten, and then dividing the results by two.

Solution symbol

Answer

Dividing 1500 by 10 gives 150. To divide 150 by 2, you can either split the problem up by dividing both 100 and 50 by two, and adding the results together to get 50 and 25, or you can write it out more formally like this:

multiline equation line 1 75 line 2 two times 150

Thus, the store should purchase 75 packs of paper. You might have used a slightly different approach to arrive at an answer—as long as your answer is correct, that’s okay!

There are lots of different strategies to help with division problems here, so don’t expect to feel comfortable with them all straight away. You might find it useful to start with to pick the first two and work on these. Good luck—and remember, if you find something that works well for you, it is fine to stick to that.

3.3.11 Calculator Exploration—A Practical Problem

Let’s return to the calculator for a practical problem.

Calculator symbol The calculator can be accessed in the left-hand side bar under Toolkit.

Try each part of the question below. Don’t panic if you can’t see how to get the answer right away. The hints will give you some direction.

Activity symbolActivity: The Great Malvern Priory

The Great Malvern Priory in England is a church dating back more than 900 years. It contains magnificent stained glass, medieval wall tiles, and beautiful carved monks’ seats. In 2011, there was a notice posted at the entry to the priory, reading: “This Priory Church costs £3 every 5 minutes.” Visitors are encouraged to leave a donation of £2.50.

(a) To maintain the priory costs £3 (three British pounds) every five minutes. Use the calculator to find out how much it costs (in British pounds) in one year (365 days).

Hint symbol
Discussion

How many times do you have “five minutes” in one hour? Use this to find the cost per hour. Now, how many hours are in a day, and how many days are in a year?

Solution symbol
Answer

(a) There are 12 stretches of five minutes in each hour, and 24 hours in each day. Therefore, the cost for a year of 365 days will be equation left hand side pound times three multiplication 12 multiplication 24 multiplication 365 equals right hand side pound times 315360. The cost of running the priory for a year is £315,360.

(b) In 2011, one British pound was equivalent to about 1.55 U.S. dollars. How much will it cost an American visitor (in U.S. dollars) when he or she donates £2.50 to the Priory?

Hint symbol
Discussion

To buy one British pound, an American had to pay $1.55. The visitor then donates £2.50.

Solution symbol
Answer

(b) Since equation left hand side 2.5 multiplication dollar times 1.55 equals right hand side dollar times 3.875, it will cost the American visitor about $3.88 (rounded to the nearest cent).

(c) Calculate how much the Priory would cost to run for a year in U.S. dollars, using the same exchange rate as before (£1 costs $1.55).

Hint symbol
Discussion

If you find this calculation difficult, then think of a simpler version first. One British pound is equivalent to 1.55 U.S. dollars, so two British pounds would give twice as many dollars—multiply by 2. So, how many dollars would you get for £315,360?

Solution symbol
Answer

(c) equation sequence pound times 315360 equals dollar times 1.55 multiplication 315360 equals dollar times 488808 So the cost in US dollars is $488,808.

(d) You found that the cost for a year is £315,360. If each visitor to the Priory leaves the suggested donation of £2.50, how many visitors are needed in a year to cover the cost? How many does that make per day?

Hint symbol
Discussion

If you’re not sure how to do this, then think of a similar problem with easier numbers. Suppose that the total cost was £100, and each visitor donated £5. To find the number of visitors necessary, you would need to divide £100 by £5. This makes 20, and you know that this is right, because 20 visitors at £5 each will give £100. Can you apply the same method to this bigger problem? When you have found the number of visitors needed per year, you can divide this by the number of days in a year to find  how many people have to donate each day in order to raise the money needed.

Solution symbol
Answer

Dividing £315,360 by £2.50 gives 126,144. So, 126,144 visitors would be needed each year to cover the total cost by paying £2.50 each. Dividing by 365, this comes out at 345.6, which is nearly 350 paying visitors that will have to contribute each day to cover the cost of maintaining the church.

Well done: you used the calculator to solve a mathematical problem from the real world!

Did you write out the workings that you used to get to your answer in your math notebook? It is good to get into this habit for a few reasons. Firstly, it makes it easier for you to check back over your work, following your own reasoning, and checking for possible errors. Also, if you are going to have your work graded or marked, then your teacher can see everything you have done.  They can then give you credit for what you have done well and be able to help out if they spot any problems. Never just put the answer down without showing how you got to it.

3.3.12 Multiplication and Division Puzzle

Multiplication and division require practice; the best way to learn is to invest time. Depending on how you learn, you might spend that time playing games, practising on paper, and/or watching videos.

Here’s a multiplication and division puzzle you can baffle your friends and family with.

Give someone your calculator and tell them “Think of any three-digit number and enter it on the calculator—for example, 562. Now enter the same three figures again to give you a six-digit number—for example, 562,562.”

Now, you pretend to think hard and finally say, “Divide that six-digit number by … er … 13.”

Pretend to think hard again, and then say, “Now, divide the number you’ve got by … um … 11.”

Finally, say, “Right—I’ve got it! Divide your new number by seven, and you’ve got back to the number you started with.” Hopefully they’ll be impressed by your new math skills!

You can tell them how it works if you want. As you now know, dividing by 13, then 11, and then seven is the reverse of multiplying by 13, then 11 then seven.

13 multiplication seven multiplication 11 equals 1001 (try it on your calculator). Multiplying any three-digit number by 1001 will give you the same three figures repeated: try that now. It’s like multiplying any single digit by 11 will give you the figure repeated: six multiplication 11 equals 66, for example, and eight multiplication 11 equals 88.

So dividing a six-digit number, made up of a repeated pair of three-digit numbers, by 13, 11, and then seven will result in returning to the number your friend started with. Have fun!

Well done, you have reached end of another unit! Before you move on try this game, which helps us practice our multiplication skills.

 Now it is time to do the self-check questions and end of unit quiz. Good luck!

3.4 Self-Check

The more you practice, the more your skills improve. Below are some exercises that will help you continue to develop your ability and check to make sure you understand the concepts discussed in this unit. Be sure to write your work out in your math notebook so that you can refer to it later if necessary.

Exercise symbolExercise 1: Adding it Up

(a) sum with, 3 , summands 24.08 plus 68.2 plus 105.59

Solution symbol

Answer

(a) multiline equation line 1 separator 21 times four full stop times 01 times eight line 2 six eight .2 line 3 separator plus plus 105.59 low line line 4 197.87

Estimate:equation sequence sum with, 3 , summands 25 plus 70 plus 105 equals sum with, 3 , summands 25 plus 105 plus 70 equals 200; this is close to the answer you calculated.

(b) 198.22 plus 32.8

Solution symbol

Answer

(b) multiline equation line 1 11 times 91 times 81 times .2 two line 2 separator plus 32.8 low line line 3 two 31 .0 two

Estimate: 200 plus 33 equals 233; this is close to the answer you calculated.

(c) sum with, 3 , summands 35.4 plus 292.56 plus 29.35

Solution symbol

Answer

(c) sum with, 3 , summands 35.4 plus 292.56 plus 29.35 equals 357.31

Estimate: 35 + 300 + 30 = 365; this is close to the answer you calculated.

(d) sum with, 3 , summands 1027 plus 2.95 plus 329.6

Solution symbol

Answer

(d) sum with, 3 , summands 1027 plus 2.95 plus 329.6 equals 1359.55

Estimate: 1030 + 3 + 330 = 1363: this is close to the answer you calculated.

Exercise symbolExercise 2: Subtracting

For each subtraction problem, calculate the difference on paper, then use an estimate to check if your answer is reasonable. Check your work by using addition.

183.72 minus 65.68

Solution symbol

Answer

(a) multiline equation line 1 one times 87 times 31 full stop 76 times 21 line 2 full stop separator minus 65 full stop separator 68 low line line 3 one one eight full stop zero four

Estimate:185 minus 65 equals 120; this is close to the answer you calculated.

Check using addition: 118.04 plus 65.68 equals 183.72

(b) 245.06 minus 20.2

Solution symbol

Answer

(b) multiline equation line 1 24 times 54 full stop 01 six line 2 full stop separator minus 20 full stop separator two low line line 3 two two four full stop eight six

Estimate: 250 minus 20 equals 230; this is close to the answer you calculated.

Check using addition: 20.2 plus 224.86 equals 245.06.

(c) 305.6 minus 90.07

Solution symbol

Answer

(c) 305.6 minus 90.07 equals 215.53

Estimate: 300 minus 90 equals 210; this is close to the answer you calculated.

Check using addition: 90.07 plus 215.53 equals 305.6.

(d) 7230 minus 321.6

Solution symbol

Answer

(d) 7230 minus 321.6 equals 6908.4

Estimate: 7200 minus 300 equals 6900 semicolon this is close to the answer you calculated.

Check using addition: 321.6 plus 6908.4 equals 7230.

Exercise symbolExercise 3: Multiplying

Carry out these calculations and then use an estimate to check whether your answer is reasonable.

(a) 422 times normal x times 38

Solution symbol

Answer

(a) multiline equation line 1 422 line 2 separator multiplication 38 low line line 3 3376 line 4 plus plus 12660 low line line 5 16036

Estimate:400 times normal x times 40 equals 16000; this is close to the answer you calculated.

Carry out these calculations and then use an estimate to check whether your answer is reasonable.

(b) 632 times normal x times 29

Solution symbol

Answer

(b) multiline equation line 1 623 line 2 separator multiplication 29 low line line 3 5688 line 4 plus plus plus 12640 low line line 5 18328

Estimate:600 times normal x times 30 equals 18000; this is close to the answer you calculated.

Carry out these calculations and then use an estimate to check whether your answer is reasonable.

(c) 584 times normal x times 72

Solution symbol

Answer

(c) multiline equation line 1 584 line 2 separator multiplication 72 low line line 3 1168 line 4 plus plus plus 40880 low line line 5 42048

Estimate:600 times normal x times 70 equals 42000; this is close to the answer you calculated.

Exercise symbolExercise 4: Dividing

Carry out these calculations and then use an estimate to check whether your answer is reasonable.

(a) 267 division 12

Solution symbol

Answer

(a) equation sequence 267 equals 20 times normal x times 12 plus 27 equals 22 times normal x times 12 plus three

Thus, equation left hand side 267 division 12 equals right hand side 22 times cap r times 13.

Estimate: 250 division 10 equals 25; this is close to our answer you calculated.

(b) 265 division 25

Solution symbol

Answer

(b) 265 equals 25 times normal x times 10 plus 15

Thus, equation left hand side 265 division 25 equals right hand side 10 times cap r times 15.

Estimate:250 division 25 equals 10; this is close to the answer you calculated.

(c) 299 division 13

Solution symbol

Answer

(c) equation sequence 299 equals 13 times normal x times 20 plus 39 equals left parenthesis 13 x 20 right parenthesis plus left parenthesis three x 13 right parenthesis equals left parenthesis 13 x 20 right parenthesis equals 13 x 23 right parenthesis

Thus, 299 division 13 equals 23.

Estimate:300 division 15 equals 20; this is close to the answer you calculated.

3.5 Quiz Time

Practice Quiz

Now that you have taken the time to work through these sections, do this short quiz! It will help you to monitor your progress, particularly if you took the quiz at the start of the unit as well.

The quiz checks most of the topics in the unit, and should give you a good 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!

Click here for the post quiz.

3.6 Study Checklist

Study Checklist

Read through the list below and think over all the work you have done in this unit. If there is a checkpoint that doesn’t seem familiar, skim your notes to jog your memory. Remember that your mathematical skills will develop and grow stronger over time:

  • Use addition, subtraction, multiplication, and division in different situations.
  • Understand different strategies for mental math problems.
  • Use estimation to check your answers.
  • Use more math language.

You’ve already completed three units, which is a great achievement—so keep up the good work!

Outlook on Unit 4

Now let’s move onto Unit 4 and continue to work with these operations. We will look at exponents, the order to carry out these operations, and some real world problems. 

Ready for Unit 4? Click here to begin!