CHAPTER 1 Whole Numbers, Integers, and Introduction to Algebra

# 1.5 Multiply and Divide Integers

Learning Objectives

By the end of this section, you will be able to:

• Multiply integers
• Divide integers
• Simplify expressions with integers
• Evaluate variable expressions with integers
• Translate English phrases to algebraic expressions
• Use integers in applications

# Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

We remember that means add a, b times. Here, we are using the model just to help us discover the pattern.

The next two examples are more interesting.

What does it mean to multiply 5 by It means subtract 5, 3 times. Looking at subtraction as “taking away,” it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away 5 three times.

In summary:

Notice that for multiplication of two signed numbers, when the:

• signs are the same, the product is positive.
• signs are different, the product is negative.

We’ll put this all together in the chart below

Multiplication of Signed Numbers

For multiplication of two signed numbers:

 Same signs Product Example Two positives Two negatives Positive Positive
 Different signs Product Example Positive \cdot negative Negative \cdot positive Negative Negative

EXAMPLE 1

Multiply: a) b) c) d) .

Solution
 a) Multiply, noting that the signs are different so the product is negative. b) Multiply, noting that the signs are the same so the product is positive. c) Multiply, with different signs. d) Multiply, with same signs.

TRY IT 1.1

Multiply: a) b) c) d) .

a) b) 28 c) d) 60

TRY IT 1.2

Multiply: a) b) c) d) .

a) b) 54 c) d) 39

When we multiply a number by 1, the result is the same number. What happens when we multiply a number by ? Let’s multiply a positive number and then a negative number by to see what we get.

Each time we multiply a number by , we get its opposite!

Multiplication by

Multiplying a number by gives its opposite.

EXAMPLE 2

Multiply: a) b) .

Solution
 a) Multiply, noting that the signs are different so the product is negative. b) Multiply, noting that the signs are the same so the product is positive.

TRY IT 2.1

Multiply: a) b) .

a) b) 17

TRY IT 2.2

Multiply: a) b) .

a) b) 16

# Divide Integers

What about division? Division is the inverse operation of multiplication. So, because . In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15. Look at some examples of multiplying integers, to figure out the rules for dividing integers.

Division follows the same rules as multiplication!

For division of two signed numbers, when the:

• signs are the same, the quotient is positive.
• signs are different, the quotient is negative.

And remember that we can always check the answer of a division problem by multiplying.

Multiplication and Division of Signed Numbers

For multiplication and division of two signed numbers:

• If the signs are the same, the result is positive.
• If the signs are different, the result is negative.
Same signs Result
Two positives Positive
Two negatives Positive

If the signs are the same, the result is positive.

Different signs Result
Positive and negative Negative
Negative and positive Negative

If the signs are different, the result is negative.

EXAMPLE 3

Divide: a) b) .

Solution
 a) Divide. With different signs, the quotient is negative. b) Divide. With signs that are the same, the quotient is positive.

TRY IT 3.1

Divide: a) b) .

a) b) 39

TRY IT 3.2

Divide: a) b) .

a) b) 23

# Simplify Expressions with Integers

What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

EXAMPLE 4

Simplify: .

Solution

TRY IT 4.1

Simplify: .

TRY IT 4.2

Simplify: .

EXAMPLE 5

Simplify: a) b) .

Solution
 a) Write in expanded form. Multiply. Multiply. Multiply. b) Write in expanded form. We are asked to find the opposite of. Multiply. Multiply. Multiply.

Notice the difference in parts a) and b). In part a), the exponent means to raise what is in the parentheses, the to the power. In part b), the exponent means to raise just the 2 to the power and then take the opposite.

TRY IT 5.1

Simplify: a) b) .

a) 81 b)

TRY IT 5.2

Simplify: a) b) .

a) 49 b)

The next example reminds us to simplify inside parentheses first.

EXAMPLE 6

Simplify: .

Solution
 Subtract in parentheses first. Multiply. Subtract.

TRY IT 6.1

Simplify: .

29

TRY IT 6.2

Simplify: .

52

EXAMPLE 7

Simplify: .

Solution
 Exponents first. Multiply. Divide.

TRY IT 7.1

Simplify: .

4

TRY IT 7.2

Simplify: .

9

EXAMPLE 8

Simplify: .

Solution
 Multiply and divide left to right, so divide first. Multiply. Add.

TRY IT 8.1

Simplify: .

21

TRY IT 8.2

Simplify: .

6

# Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

EXAMPLE 9

When , evaluate: a) b) .

Solution

a)

 Simplify. −4

b)

TRY IT 9.1

When , evaluate a) b) .

a) b) 10

TRY IT 9.2

When , evaluate a) b) .

a) b) 17

EXAMPLE 10

Evaluate when and .

Solution
 Add inside parenthesis. (6)2 Simplify. 36

TRY IT 10.1

Evaluate when and .

196

TRY IT 10.2

Evaluate when and .

8

EXAMPLE 11

Evaluate when a) and b) .

Solution

a)

 Subtract. 8

b)

 Subtract. 32

TRY IT 11.1

Evaluate: when a) and b) .

a) b) 36

TRY IT 11.2

Evaluate: when a) and b) .

a) b) 9

EXAMPLE 12

Evaluate: when .

Solution

Substitute . Use parentheses to show multiplication.

 Substitute. Evaluate exponents. Multiply. Add. 52

TRY IT 12.1

Evaluate: when .

39

TRY IT 12.2

Evaluate: when .

13

# Translate Phrases to Expressions with Integers

Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

EXAMPLE 13

Translate and simplify: the sum of 8 and , increased by 3

Solution
 the sum of 8 and , increased by 3. Translate. Simplify. Be careful not to confuse the brackets with an absolute value sign. Add.

TRY IT 13.1

Translate and simplify the sum of 9 and , increased by 4

TRY IT 13.2

Translate and simplify the sum of and , increased by 7

When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

minus
the difference of and
subtracted from
less than

Be careful to get a and b in the right order!

EXAMPLE 14

Translate and then simplify a) the difference of 13 and b) subtract 24 from .

Solution
 a) Translate. Simplify. b) Translate. Remember, “subtract from means . Simplify.

TRY IT 14.1

Translate and simplify a) the difference of 14 and b) subtract 21 from .

a) b)

TRY IT 14.2

Translate and simplify a) the difference of 11 and b) subtract 18 from .

a) b)

Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”

EXAMPLE 15

Translate to an algebraic expression and simplify if possible: the product of and 14

Solution
 Translate. Simplify.

TRY IT 15.1

Translate to an algebraic expression and simplify if possible: the product of and 12

TRY IT 15.2

Translate to an algebraic expression and simplify if possible: the product of 8 and .

EXAMPLE 16

Translate to an algebraic expression and simplify if possible: the quotient of and .

Solution
 Translate. Simplify.

TRY IT 16.1

Translate to an algebraic expression and simplify if possible: the quotient of and .

TRY IT 16.2

Translate to an algebraic expression and simplify if possible: the quotient of and .

# Use Integers in Applications

We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

EXAMPLE 17

The temperature in Sparwood, British Columbia, one morning was 11 degrees. By mid-afternoon, the temperature had dropped to degrees. What was the difference of the morning and afternoon temperatures?

Solution

TRY IT 17.1

The temperature in Whitehorse, Yukon, one morning was 15 degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?

The difference in temperatures was 45 degrees.

TRY IT 17.2

The temperature in Quesnel, BC, was degrees at lunchtime. By sunset the temperature had dropped to degrees. What was the difference in the lunchtime and sunset temperatures?

The difference in temperatures was 9 degrees.

HOW TO: Apply a Strategy to Solve Applications with Integers

1. Read the problem. Make sure all the words and ideas are understood
2. Identify what we are asked to find.
3. Write a phrase that gives the information to find it.
4. Translate the phrase to an expression.
5. Simplify the expression.
6. Answer the question with a complete sentence.

EXAMPLE 18

The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

Solution
 Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are asked to find. the number of yards lost Step 3. Write a phrase that gives the information to find it. three times a 15-yard penalty Step 4. Translate the phrase to an expression. Step 5. Simplify the expression. Step 6. Answer the question with a complete sentence. The team lost 45 yards.

TRY IT 18.1

The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of 15 yards. What is the number of yards lost due to penalties?

The Bears lost 105 yards.

TRY IT 18.2

Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM? Show answer A$16 fee was deducted from his checking account.

# Key Concepts

• Multiplication and Division of Two Signed Numbers
• Same signs—Product is positive
• Different signs—Product is negative
• Strategy for Applications
1. Identify what you are asked to find.
2. Write a phrase that gives the information to find it.
3. Translate the phrase to an expression.
4. Simplify the expression.
5. Answer the question with a complete sentence.

# Practice Makes Perfect

## Multiply Integers

In the following exercises, multiply.

 1 2 3 4 5 6 7 8

## Divide Integers

In the following exercises, divide.

 9 10 11 12 13 14

## Simplify Expressions with Integers

In the following exercises, simplify each expression.

 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

## Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

 33. when a) b) 34. when a) b) 35. a) when b) when 36. a) when b) when 37.  when 38. when 39. when 40. when 41. when 42. when 43. when a) b) 44. when a) b) 45. when a) b) 46. when a) b) 47. when 48. when 49. when 50. when

## Translate English Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.

 51. the sum of 3 and , increased by 7 52. the sum of and , increased by 23 53. the difference of 10 and 54. subtract 11 from 55. the difference of and 56. subtract from 57. the product of and 58. the product of and 59. the quotient of and 60. the quotient of and 61. the quotient of and the sum of a and b 62. the quotient of and the sum of m and n 63. the product of and the difference of 64. the product of and the difference of

## Use Integers in Applications

In the following exercises, solve.

 65. Temperature On January , the high temperature in Lytton, British Columbia, was ° . That same day, the high temperature in Fort Nelson, British Columbia was °. What was the difference between the temperature in Lytton and the temperature in Embarrass? 66. Temperature On January , the high temperature in Palm Springs, California, was °, and the high temperature in Whitefield, New Hampshire was °. What was the difference between the temperature in Palm Springs and the temperature in Whitefield? 67. Football At the first down, the Chargers had the ball on their 25 yard line. On the next three downs, they lost 6 yards, gained 10 yards, and lost 8 yards. What was the yard line at the end of the fourth down? 68. Football At the first down, the Steelers had the ball on their 30 yard line. On the next three downs, they gained 9 yards, lost 14 yards, and lost 2 yards. What was the yard line at the end of the fourth down? 69. Checking Account Ester has $124 in her checking account. She writes a check for$152. What is the new balance in her checking account? 70. Checking Account Selina has $165 in her checking account. She writes a check for$207. What is the new balance in her checking account? 71. Checking Account Kevin has a balance of in his checking account. He deposits $225 to the account. What is the new balance? 72. Checking Account Reymonte has a balance of in his checking account. He deposits$281 to the account. What is the new balance?