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) .
| 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) .
Show answer
a) b) 28 c) d) 60
TRY IT 1.2
Multiply: a) b) c) d) .
Show answer
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) .
| 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) .
Show answer
a) b) 17
TRY IT 2.2
Multiply: a) b) .
Show answer
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) .
| 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) .
Show answer
a) b) 39
TRY IT 3.2
Divide: a) b) .
Show answer
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: .
| Multiply first. | |
| Add. | |
| Subtract. |
TRY IT 4.1
Simplify: .
Show answer
TRY IT 4.2
Simplify: .
Show answer
EXAMPLE 5
Simplify: a) b) .
| 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) .
Show answer
a) 81 b)
TRY IT 5.2
Simplify: a) b) .
Show answer
a) 49 b)
The next example reminds us to simplify inside parentheses first.
EXAMPLE 6
Simplify: .
| Subtract in parentheses first. | |
| Multiply. | |
| Subtract. |
TRY IT 6.1
Simplify: .
Show answer
29
TRY IT 6.2
Simplify: .
Show answer
52
EXAMPLE 7
Simplify: .
| Exponents first. | |
| Multiply. | |
| Divide. |
TRY IT 7.1
Simplify: .
Show answer
4
TRY IT 7.2
Simplify: .
Show answer
9
EXAMPLE 8
Simplify: .
| Multiply and divide left to right, so divide first. | |
| Multiply. | |
| Add. |
TRY IT 8.1
Simplify: .
Show answer
21
TRY IT 8.2
Simplify: .
Show answer
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) .
a)
| Simplify. | −4 |
b)
| Simplify. | |
| Add. | 6 |
TRY IT 9.1
When , evaluate a) b) .
Show answer
a) b) 10
TRY IT 9.2
When , evaluate a) b) .
Show answer
a) b) 17
EXAMPLE 10
Evaluate when and .
| Add inside parenthesis. | (6)2 |
| Simplify. | 36 |
TRY IT 10.1
Evaluate when and .
Show answer
196
TRY IT 10.2
Evaluate when and .
Show answer
8
EXAMPLE 11
Evaluate when a) and b) .
a)
| Subtract. | 8 |
b)
| Subtract. | 32 |
TRY IT 11.1
Evaluate: when a) and b) .
Show answer
a) b) 36
TRY IT 11.2
Evaluate: when a) and b) .
Show answer
a) b) 9
EXAMPLE 12
Evaluate: when .
Substitute . Use parentheses to show multiplication.
| Substitute. | |
| Evaluate exponents. | |
| Multiply. | |
| Add. | 52 |
TRY IT 12.1
Evaluate: when .
Show answer
39
TRY IT 12.2
Evaluate: when .
Show answer
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
| 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
Show answer
TRY IT 13.2
Translate and simplify the sum of and , increased by 7
Show answer
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 .
| 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 .
Show answer
a) b)
TRY IT 14.2
Translate and simplify a) the difference of 11 and b) subtract 18 from .
Show answer
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
| Translate. | |
| Simplify. |
TRY IT 15.1
Translate to an algebraic expression and simplify if possible: the product of and 12
Show answer
TRY IT 15.2
Translate to an algebraic expression and simplify if possible: the product of 8 and .
Show answer
EXAMPLE 16
Translate to an algebraic expression and simplify if possible: the quotient of and .
| Translate. | |
| Simplify. |
TRY IT 16.1
Translate to an algebraic expression and simplify if possible: the quotient of and .
Show answer
TRY IT 16.2
Translate to an algebraic expression and simplify if possible: the quotient of and .
Show answer
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?
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?
Show answer
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?
Show answer
The difference in temperatures was 9 degrees.
HOW TO: Apply a Strategy to Solve Applications with Integers
- Read the problem. Make sure all the words and ideas are understood
- Identify what we are asked to find.
- Write a phrase that gives the information to find it.
- Translate the phrase to an expression.
- Simplify the expression.
- 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?
| 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?
Show answer
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
- Identify what you are asked to find.
- Write a phrase that gives the information to find it.
- Translate the phrase to an expression.
- Simplify the expression.
- 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? |
Everyday Math
| 73. Stock market Javier owns 300 shares of stock in one company. On Tuesday, the stock price dropped $12 per share. What was the total effect on Javier’s portfolio? | 74. Weight loss In the first week of a diet program, eight women lost an average of 3 pounds each. What was the total weight change for the eight women? |
Writing Exercises
| 75. In your own words, state the rules for multiplying integers. | 76. In your own words, state the rules for dividing integers. |
| 77. Why is | 78. Why is |
Answers
| 1. | 3. | 5. |
| 7. 14 | 9. | 11. 13 |
| 13. | 15. | 17. 64 |
| 19. | 21. 90 | 23. 9 |
| 25. 41 | 27. | 29. |
| 31. 5 | 33. a) b) 16 | 35. a) b) 10 |
| 37. | 39. | 41. 121 |
| 43. a) 1 b) 33 | 45. a)b) 25 | 47. 21 |
| 49. | 51. | 53. |
| 55. | 57. | 59. |
| 61. | 63. | 65. ° |
| 67. 21 | 69. | 71. $187 |
| 73. | 75. Answers may vary | 77. Answers may vary |
Attributions
This chapter has been adapted from “Multiply and Divide Integers” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.