1.4 Multiply and Divide Integers

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 $a \cdot b$ means add a, b times.

Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads “add 5, 3 times.” Below this depicts three rows of blue counters, with five counters in each row. Under this, it says “15 positives.” Under thisis the equation“5 times 3 equals 15.” The image on the right reads “negative 5 times three. The three is in parentheses. Below this it reads, “add negative five, three times.” Under this are fifteen red counters in three rows of five. Below this it reads” “15 negatives”. Below this is the equation negative five times 3 equals negative 15.”

The next two examples are more interesting.

What does it mean to multiply 5 by $−3 ?$ 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.

This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.

In summary:

$\begin{matrix} 5 \cdot 3 & = & 15 & −5 (3) & = & −15 \\ 5 (−3) & = & −15 & (−5) (−3) & = & 15 \end{matrix}$

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.

Note

Multiplication of Signed Numbers

For multiplication of two signed numbers:

Same signs	Product	Example
Two positives Two negatives	Positive Positive	$\begin{matrix} 7 \cdot 4 & = & 28 \\ −8 (−6) & = & 48 \end{matrix}$

Different signs	Product	Example
Positive · negative Negative · positive	Negative Negative	$\begin{matrix} 7 (−9) & = & −63 \\ −5 \cdot 10 & = & −50 \end{matrix}$

Example 1

Multiply: ⓐ $−9 times 3$ ⓑ $−2 open parenthesis −5 close parenthesis$ ⓒ $4 open parenthesis −8 close parenthesis$ ⓓ $7 times 6.$

**Table 1.8**
ⓐ Multiply, noting that the signs are different so the product is negative.	$−9 times 3; −27$
ⓑ Multiply, noting that the signs are the same so the product is positive.	$−2 open parenthesis −5 close parenthesis; 10$
ⓒ Multiply, with different signs.	$4 open parenthesis −8 close parenthesis; −32$
ⓓ Multiply, with same signs.	$7 times 6; 42$

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

$\begin{matrix} −1 \cdot 4 & −1 (−3) \\ Multiply. & −4 & 3 \\ −4 is the opposite of 4 . & 3 is the opposite of −3 . \end{matrix}$

Each time we multiply a number by $−1,$ we get its opposite!

Example 2

Multiply: ⓐ $−1 times 7$ ⓑ $−1 open parenthesis −11 close parenthesis.$

**Table 1.9**
ⓐ Multiply, noting that the signs are different so the product is negative.	$\begin{matrix} −1 \cdot 7 \\ −7 \\ −7 is the opposite of 7 . \end{matrix}$
ⓑ Multiply, noting that the signs are the same so the product is positive.	$−1 open parenthesis −11 close parenthesis; 11; 11 is the opposite of −11.$

Divide Integers

What about division? Division is the inverse operation of multiplication. So, $15 \div 3 = 5$ because $5 \cdot 3 = 15 .$ 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.

$\begin{matrix} 5 \cdot 3 & = & 15 so 15 \div 3 & = & 5 & −5 (3) & = & −15 so −15 \div 3 & = & −5 \\ (−5) (−3) & = & 15 so 15 \div (−3) & = & −5 & 5 (−3) & = & −15 so −15 \div (−3) & = & 5 \end{matrix}$

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.

Note

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 Two negatives	Positive Positive
If the signs are the same, the result is positive.

Different signs	Result
Positive and negative Negative and positive	Negative Negative
If the signs are different, the result is negative.

Example 3

Divide: ⓐ $−27 divided by 3$ ⓑ $−100 divided by open parenthesis −4 close parenthesis.$

**Table 1.10**
ⓐ Divide. With different signs, the quotient is negative.	$\begin{matrix} - 27 \div 3 \\ −9 \end{matrix}$
ⓑ Divide. With signs that are the same, the quotient is positive.	$\begin{matrix} - 100 \div (−4) \\ 25 \end{matrix}$

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 5

Simplify: ⓐ $open parenthesis −2 close parenthesis to the power 4$ ⓑ $− 2 to the power 4.$

**Table 1.12**
ⓐ Write in expanded form. Multiply. Multiply. Multiply.	$open parenthesis −2 close parenthesis to the power 4; open parenthesis −2 close parenthesis open parenthesis −2 close parenthesis open parenthesis −2 close parenthesis open parenthesis −2 close parenthesis; 4 open parenthesis −2 close parenthesis open parenthesis −2 close parenthesis; −8 open parenthesis −2 close parenthesis; 16$
ⓑ Write in expanded form. We are asked to find the opposite of $2 to the power 4.$ Multiply. Multiply. Multiply.	$\begin{matrix} − 2^{4} \\ − (2 \cdot 2 \cdot 2 \cdot 2) \\ − (4 \cdot 2 \cdot 2) \\ − (8 \cdot 2) \\ -16 \end{matrix}$

Notice the difference in parts ⓐ and ⓑ. In part ⓐ, the exponent means to raise what is in the parentheses, the $open parenthesis −2 close parenthesis$ to the $4 to the power th$ power. In part ⓑ, the exponent means to raise just the 2 to the $4 to the power th$ power and then take the opposite.

The next example reminds us to simplify inside parentheses first.

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.

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 $−12,$ increased by 3.

**Table 1.16**
	the sum of 8 and $−12,$ increased by 3.
Translate.	$[8 + (−12)] + 3$
Simplify. Be careful not to confuse the brackets with an absolute value sign.	$open parenthesis −4 close parenthesis plus 3$
Add.	$−1$

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.

$a - b$
$a$ minus $b$ the difference of $a$ and $b$ $b$ subtracted from $a$ $b$ less than $a$

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

Example 14

Translate and then simplify ⓐ the difference of 13 and $−21$ ⓑ subtract 24 from $−19.$

**Table 1.17**
ⓐ Translate. Simplify.	$\begin{matrix} the difference of 13 and - 21 \\ 13 - (−21) \\ 34 \end{matrix}$
ⓑ Translate. Remember, "subtract $b$ from $a$ means $a - b$ . Simplify.	$\begin{matrix} subtract 24 from - 19 \\ - 19 - 24 \\ - 43 \end{matrix}$

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.”

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 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?

**Table 1.20**
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.	$3 open parenthesis −15 close parenthesis$
Step 5. Simplify the expression.	$−45$
Step 6. Answer the question with a complete sentence.	The team lost 45 yards.

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.

$−4 times 8$

$−32$

$−3 times 9$

$9 open parenthesis −7 close parenthesis$

$−63$

$13 open parenthesis −5 close parenthesis$

$- 1 \cdot 6$

$−6$

$- 1 \cdot 3$

$−1 open parenthesis −14 close parenthesis$

$−1 open parenthesis −19 close parenthesis$

Divide Integers

In the following exercises, divide.

$−24 divided by 6$

$−4$

$35 divided by open parenthesis −7 close parenthesis$

$−52 divided by open parenthesis −4 close parenthesis$

$−84 divided by open parenthesis −6 close parenthesis$

$−180 divided by 15$

$−12$

$−192 divided by 12$

Simplify Expressions with Integers

In the following exercises, simplify each expression.

$5 (−6) + 7 (−2) - 3$

$−47$

$8 (−4) + 5 (−4) - 6$

$open parenthesis −2 close parenthesis to the power 6$

$open parenthesis −3 close parenthesis to the power 5$

$− 4 squared$

$−16$

$− 6 squared$

$−3 open parenthesis −5 close parenthesis open parenthesis 6 close parenthesis$

$−4 open parenthesis −6 close parenthesis open parenthesis 3 close parenthesis$

$(8 - 11) (9 - 12)$

$(6 - 11) (8 - 13)$

$26 - 3 (2 - 7)$

$23 - 2 (4 - 6)$

$65 \div (−5) + (−28) \div (−7)$

$−9$

$52 \div (−4) + (−32) \div (−8)$

$9 - 2 [3 - 8 (−2)]$

$−29$

$11 - 3 [7 - 4 (−2)]$

${(−3)}^{2} - 24 \div (8 - 2)$

${(−4)}^{2} - 32 \div (12 - 4)$

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

$y plus open parenthesis −14 close parenthesis$ when ⓐ $y equals −33$ ⓑ $y equals 30$

ⓐ $−47$ ⓑ 16

$x plus open parenthesis −21 close parenthesis$ whenⓐ $x equals −27$ ⓑ $x equals 44$

ⓐ $a plus 3$ when $a equals −7$
ⓑ $− a plus 3$ when $a equals −7$

ⓐ $−4$ ⓑ 10

ⓐ $d plus open parenthesis −9 close parenthesis$ when $d equals −8$
ⓑ $− d plus open parenthesis −9 close parenthesis$ when $d equals −8$

$m + n$ when
$m = −15, n = 7$

$−8$

$p + q$ when
$p = −9, q = 17$

$r + s$ when $r = −9, s = −7$

$−16$

$t + u$ when $t = −6, u = −5$

${(x + y)}^{2}$ when
$x = −3, y = 14$

121

${(y + z)}^{2}$ when
$y = −3, z = 15$

$−2 x plus 17$ when

ⓐ $x equals 8$
ⓑ $x equals −8$

ⓐ 1
ⓑ 33

$−5 y plus 14$ when
ⓐ $y equals 9$
ⓑ $y equals −9$

$10 minus 3 m$ when
ⓐ $m equals 5$
ⓑ $m equals −5$

ⓐ $−5$ ⓑ 25

$18 minus 4 n$ when
ⓐ $n equals 3$
ⓑ $n equals −3$

$2 w^{2} - 3 w + 7$ when
$w equals −2$

$3 u^{2} - 4 u + 5$ when $u equals −3$

$9 a - 2 b - 8$ when
$a = −6 and b = −3$

$−56$

$7 m - 4 n - 2$ when
$m = −4 and n = −9$

Translate English Phrases to Algebraic Expressions

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

the sum of 3 and $−15,$ increased by 7

$(3 + (−15)) + 7; - 5$

the sum of $−8$ and $−9,$ increased by 23

the difference of 10 and $−18$

$10 minus open parenthesis −18 close parenthesis; 28$

subtract 11 from $−25$

the difference of $−5$ and $−30$

$−5 minus open parenthesis −30 close parenthesis; 25$

subtract $−6$ from $−13$

the product of $−3 and 15$

$−3 \cdot 15; - 45$

the product of $−4 and 16$

the quotient of $−60$ and $−20$

$−60 divided by open parenthesis −20 close parenthesis; 3$

the quotient of $−40$ and $−20$

the quotient of $−6$ and the sum of a and b

$\frac{−6}{a + b}$

the quotient of $−7$ and the sum of m and n

the product of $−10$ and the difference of $p and q$

$−10 open parenthesis p minus q close parenthesis$

the product of $−13$ and the difference of $c and d$

Use Integers in Applications

In the following exercises, solve.

Temperature On January $15,$ the high temperature in Anaheim, California, was $84 °.$ That same day, the high temperature in Embarrass, Minnesota was $−12 °.$ What was the difference between the temperature in Anaheim and the temperature in Embarrass?

$96 °$

Temperature On January $21,$ the high temperature in Palm Springs, California, was $89 °,$ and the high temperature in Whitefield, New Hampshire was $−31 °.$ What was the difference between the temperature in Palm Springs and the temperature in Whitefield?

Football On the first down, the Chargers had the ball on their 25-yard line. They lost 6 yards on the first-down play, gained 10 yards on the second-down play, and lost 8 yards on the third-down play. What was the yard line at the end of the third-down play?

Football On first down, the Steelers had the ball on their 30-yard line. They gained 9 yards on the first-down play, lost 14 yards on the second-down play, and lost 2 yards on the third-down play. What was the yard line at the end of the third-down play?

Checking Account Mayra has $124 in her checking account. She writes a check for $152. What is the new balance in her checking account?

$− $28$

Checking Account Selina has $165 in her checking account. She writes a check for $207. What is the new balance in her checking account?

Checking Account Diontre has a balance of $− $38$ in his checking account. He deposits $225 to the account. What is the new balance?

$187

Checking Account Reymonte has a balance of $− $49$ in his checking account. He deposits $281 to the account. What is the new balance?

Everyday Math

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?

$− $3600$

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

In your own words, state the rules for multiplying integers.

Answers may vary

In your own words, state the rules for dividing integers.

Why is $− 2^{4} \neq {(−2)}^{4} ?$

Answers may vary

Why is $− 4 cubed equals open parenthesis −4 close parenthesis cubed ?$

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

A table is shown that is composed of four columns and seven rows. The titles of the columns are “I can …”, “Confidently”, “With some help” and “No – I don’t get it!”. The first column reads “multiple integers.”, “divide integers.”, “simplify expressions with integers.”, “evaluate variable expressions with integers.”, “translate English phrases to algebraic expressions.” and “use integers in applications.”

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

	$7 (−2) + 4 (−7) - 6$
Multiply first.	$−14 + (−28) - 6$
Add.	$−42 minus 6$
Subtract.	$−48$

	$12 - 3 (9 - 12)$
Subtract in parentheses first.	$12 minus 3 open parenthesis −3 close parenthesis$
Multiply.	$12 minus open parenthesis −9 close parenthesis$
Subtract.	$21$

	$8 open parenthesis −9 close parenthesis divided by open parenthesis −2 close parenthesis cubed$
Exponents first.	$8 open parenthesis −9 close parenthesis divided by open parenthesis −8 close parenthesis$
Multiply.	$−72 divided by open parenthesis −8 close parenthesis$
Divide.	$9$

	$−30 \div 2 + (−3) (−7)$
Multiply and divide left to right, so divide first.	$−15 plus open parenthesis −3 close parenthesis open parenthesis −7 close parenthesis$
Multiply.	$−15 plus 21$
Add.	$6$

	$the product of −2 and 14$
Translate.	$open parenthesis −2 close parenthesis open parenthesis 14 close parenthesis$
Simplify.	$−28$

	$the quotient of −56 and −7$
Translate.	$−56 divided by open parenthesis −7 close parenthesis$
Simplify.	$8$



Add inside parenthesis.	(6)²
Simplify.	36