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# 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!

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:

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

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.

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 gives the same answer as . In mathematical terms, we say that addition is commutative. Not every operation will possess this property (for example ). 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 : 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: 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. ###### 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. ###### 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 and 37 into . This makes the sum much easier to work out in your head and the sum then becomes . 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

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

(a)

###### Discussion

Consider breaking down numbers to make easy addition problems.

For example,. Splitting the numbers up like this means that you can work in tens, which makes them much easier to add up in your head.

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

Or, .

(b)

(b)

(c)

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

Alternatively, you could write .

(d)

(d) .

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

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 . This is the same answer as the one obtained by subtraction: . 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 , 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: 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?

###### Discussion

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

After giving your friend $10, you will still owe her . You still have five$1 bills, so you fork those over, too! Now you only owe her .

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 . After giving your friend eight dimes, the amount you still owe is .

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.

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: Subtraction Carry out the follow subtractions on paper. (a) 526 – 327 ###### Answer (a) 526 – 327 = 199 (b)1029 – 822 ###### Answer (b) 1029 – 822 = 207 (c) 776 – 389 ###### Answer (c) 776 – 389 = 387 (d) 429 – 338 ###### 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: 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? #### 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. #### 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 subtract15.75 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.

## 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.

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.

### 3.3.1 Multiplication and Division Notation

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

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

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 . 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, .

Now, say we have . 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, .

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, , because 5 goes into 8 one time, which gives us 3 left over.

We can also relate division to multiplication. We say because . 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 . Let’s say . This is equivalent to saying that , but 0 times any number results in 0, so 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, , because Zero can be a dividend or a quotient—just not a divisor.

#### Activity: Find the Quotient

(a) What is the quotient of ?

##### Discussion

Use multiplication. For example, .

a.

A. 27

b.

B. 10

c.

C. 6

d.

D. 9

(a) Because , the correct answer is 10, which is letter B.

(b) What is the quotient of ?

a.

A. 8

b.

B. 10

c.

C. 3

d.

D. 2

(b) Because , 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 . 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, is telling us to add 2 to itself 3 times: .

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

Let’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: Multiplication and Division

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

(a)

##### 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 for the division symbol.

(a)

Thus, the answer is 12,876. (This seems reasonable since .)

(b)

(b)

Thus, the answer is 12,880. (This seems reasonable since.)

(c)

(c)

So the solution is 207. (This seems reasonable because .)

(d)

(d)

So the solution is 269.

If 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 is the same as . But, what about multiplication and division? Is the same as ? Is the same as ? 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, is the same as . 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 .

You can also check it by using the idea that multiplication is repeated addition. Remember that , and . 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: is not the same as . 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 , 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 . 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 times, with three left over. To check that, multiply , and . 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 , you could think: “How can I break 159 apart to make this an easier problem to solve?”

You might think of this as:

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 , 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 .

Thus, .

### 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 , you may know off the top of your head that , 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, , which is three less than 159, so .

##### 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?

###### Discussion

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

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.

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:

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

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: 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).

##### 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?

(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 . 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?

(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). ##### 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? ##### Answer (c) 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?

##### 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.

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.

(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: , for example, and .

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 1: Adding it Up

(a)

(a)

Estimate:; this is close to the answer you calculated.

(b)

(b)

Estimate: ; this is close to the answer you calculated.

(c)

(c)

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

(d)

(d)

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

### Exercise 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.

(a)

Estimate:; this is close to the answer you calculated.

(b)

(b)

Estimate: ; this is close to the answer you calculated.

(c)

(c)

Estimate: ; this is close to the answer you calculated.

(d)

(d)

Estimate: this is close to the answer you calculated.

### Exercise 3: Multiplying

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

(a)

(a)

Estimate:; 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)

(b)

Estimate:; 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)

(c)

Estimate:; this is close to the answer you calculated.

### Exercise 4: Dividing

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

(a)

(a)

Thus, .

Estimate: ; this is close to our answer you calculated.

(b)

(b)

Thus, .

Estimate:; this is close to the answer you calculated.

(c)

(c)

Thus, .

Estimate:; 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!

## 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.