Multiply and Divide Integers — Elementary Algebra 2e

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.

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	$\begin{matrix} 7 (−9) & = & −63 \\ −5 \cdot 10 & = & −50 \end{matrix}$

Multiply: ⓐ $−9 \cdot 3$ ⓑ $−2 (−5)$ ⓒ $4 (−8)$ ⓓ $7 \cdot 6 .$

Solution

Examples of integer multiplication, illustrating how the signs of numbers determine the product's sign.
ⓐ Multiply, noting that the signs are different so the product is negative.	$\begin{matrix} −9 \cdot 3 \\ −27 \end{matrix}$
ⓑ Multiply, noting that the signs are the same so the product is positive.	$\begin{matrix} −2 (−5) \\ 10 \end{matrix}$
ⓒ Multiply, with different signs.	$\begin{matrix} 4 (−8) \\ −32 \end{matrix}$
ⓓ Multiply, with same signs.	$\begin{matrix} 7 \cdot 6 \\ 42 \end{matrix}$

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!

Multiply: ⓐ $−1 \cdot 7$ ⓑ $−1 (−11) .$

Solution

Examples demonstrating multiplication by -1, showing how the product's sign is determined and always results in the opposite value of the original number.
ⓐ 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.	$\begin{matrix} −1 (−11) \\ 11 \\ 11 is the opposite of −11 . \end{matrix}$

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.

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.

Divide: ⓐ $−27 \div 3$ ⓑ $−100 \div (−4) .$

Solution

Examples demonstrating division rules for integers with different and same signs.
ⓐ 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.

Simplify: $7 (−2) + 4 (−7) - 6 .$

Solution

Step-by-step evaluation of the mathematical expression 7(-2) + 4(-7) - 6, showing the progression of operations.
	$7 (−2) + 4 (−7) - 6$
Multiply first.	$−14 + (−28) - 6$
Add.	$−42 - 6$
Subtract.	$−48$

Simplify: ⓐ ${(−2)}^{4}$ ⓑ $− 2^{4} .$

Solution

Step-by-step evaluation of exponential expressions, differentiating between a negative base raised to a power and the negative of a positive base.
ⓐ Write in expanded form. Multiply. Multiply. Multiply.	$\begin{matrix} {(−2)}^{4} \\ (−2) (−2) (−2) (−2) \\ 4 (−2) (−2) \\ −8 (−2) \\ 16 \end{matrix}$
ⓑ Write in expanded form. We are asked to find the opposite of $2^{4} .$ 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 $(−2)$ to the $4^{th}$ power. In part ⓑ , the exponent means to raise just the 2 to the $4^{th}$ power and then take the opposite.

The next example reminds us to simplify inside parentheses first.

Simplify: $12 - 3 (9 - 12) .$

Solution

Step-by-step evaluation of a mathematical expression.
	$12 - 3 (9 - 12)$
Subtract in parentheses first.	$12 - 3 (−3)$
Multiply.	$12 - (−9)$
Subtract.	$21$

Simplify: $8 (−9) \div {(−2)}^{3} .$

Solution

Demonstrates the step-by-step evaluation of the mathematical expression 8(-9) ÷ (-2)^3, illustrating the order of operations.
	$8 (−9) \div {(−2)}^{3}$
Exponents first.	$8 (−9) \div (−8)$
Multiply.	$−72 \div (−8)$
Divide.	$9$

Simplify: $−30 \div 2 + (−3) (−7) .$

Solution

Step-by-step evaluation of a mathematical expression, illustrating the application of the order of operations to reach the final numerical result.
	$−30 \div 2 + (−3) (−7)$
Multiply and divide left to right, so divide first.	$−15 + (−3) (−7)$
Multiply.	$−15 + 21$
Add.	$6$

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.

When $n = −5,$ evaluate: ⓐ $n + 1$ ⓑ $− n + 1 .$

Solution

ⓐ



Simplify.	−4

ⓑ



Simplify.
Add.	6

Evaluate ${(x + y)}^{2}$ when $x = −18$ and $y = 24 .$

Solution



Add inside parenthesis.	(6)²
Simplify.	36

Evaluate $20 - z$ when ⓐ $z = 12$ and ⓑ $z = −12 .$

Solution

ⓐ



Subtract.	8

ⓑ



Subtract.	32

Evaluate: $2 x^{2} + 3 x + 8$ when $x = 4 .$

Solution

Substitute

4 for x .

Use parentheses to show multiplication.


Substitute.
Evaluate exponents.
Multiply.
Add.	52

Translate Phrases to Expressions with Integers

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

Translate and simplify: the sum of 8 and $−12,$ increased by 3.

Solution

Step-by-step translation and solution of a mathematical word problem involving sums and integers.
	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.	$(−4) + 3$
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!

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

Solution

Demonstrates translating verbal math problems involving integers into expressions and finding their solutions.
ⓐ 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.”

Translate to an algebraic expression and simplify if possible: the product of $−2$ and 14.

Solution

Demonstrates translation and simplification of a mathematical product from text to numerical result.
	$the product of −2 and 14$
Translate.	$(−2) (14)$
Simplify.	$−28$

Translate to an algebraic expression and simplify if possible: the quotient of $−56$ and $−7 .$

Solution

Steps to translate a verbal phrase into a mathematical expression and simplify the quotient of two negative integers.
	$the quotient of −56 and −7$
Translate.	$−56 \div (−7)$
Simplify.	$8$

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.

How to Apply a Strategy to Solve Applications with Integers

In the morning, the temperature in Urbana, Illinois was 11 degrees. By mid-afternoon, the temperature had dropped to $−9$ degrees. What was the difference of the morning and afternoon temperatures?

Solution

This is a table with two columns. The left column includes steps to solve the problem. The right column includes the math to solve the problem. In the first row, the left column says “Step 1. Read the problem. Make sure all the words and ideas are understood.” The right column is blank.

In the second row, the left column says “Step 2. Identify what we are asked to find”. The right column says, “the difference of the morning and afternoon temperatures.”

In the third row, the left column says, “Step 3. Write a phrase that gives the information to find it.” Next to this in the right column, it says “the difference of 11 and negative 9.”

In the fourth row, the left column says, “Step 4. Translate the phrase to an expression.” The right column contains 11 minus negative 9.

In the fifth row, the left column says, “Step 5. Simplify the expression.” The right column contains 20.

The final row says, “Step five. Write a complete sentence that answers the question.” Next to this in the right column, it says “the difference in temperatures was 20 degrees.”

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

Steps for solving a word problem involving a 15-yard penalty, leading to a total loss of 45 yards.
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.	$3 (−15)$
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.

$−4 \cdot 8$

Solution

$−32$

$−3 \cdot 9$

$9 (−7)$

Solution

$−63$

$13 (−5)$

$- 1 \cdot 6$

Solution

$−6$

$- 1 \cdot 3$

$−1 (−14)$

Solution

14

$−1 (−19)$

Divide Integers

In the following exercises, divide.

$−24 \div 6$

Solution

$−4$

$35 \div (−7)$

$−52 \div (−4)$

Solution

13

$−84 \div (−6)$

$−180 \div 15$

Solution

$−12$

$−192 \div 12$

Simplify Expressions with Integers

In the following exercises, simplify each expression.

$5 (−6) + 7 (−2) - 3$

Solution

$−47$

$8 (−4) + 5 (−4) - 6$

${(−2)}^{6}$

Solution

64

${(−3)}^{5}$

$− 4^{2}$

Solution

$−16$

$− 6^{2}$

$−3 (−5) (6)$

Solution

90

$−4 (−6) (3)$

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

Solution

9

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

$26 - 3 (2 - 7)$

Solution

41

$23 - 2 (4 - 6)$

$65 \div (−5) + (−28) \div (−7)$

Solution

$−9$

$52 \div (−4) + (−32) \div (−8)$

$9 - 2 [3 - 8 (−2)]$

Solution

$−29$

$11 - 3 [7 - 4 (−2)]$

${(−3)}^{2} - 24 \div (8 - 2)$

Solution

5

${(−4)}^{2} - 32 \div (12 - 4)$

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

$y + (−14)$ when ⓐ $y = −33$ ⓑ $y = 30$

Solution

ⓐ $−47$ ⓑ 16

$x + (−21)$ whenⓐ $x = −27$ ⓑ $x = 44$

ⓐ $a + 3$ when $a = −7$
ⓑ $− a + 3$ when $a = −7$

Solution

ⓐ $−4$ ⓑ 10

ⓐ $d + (−9)$ when $d = −8$
ⓑ $− d + (−9)$ when $d = −8$

$m + n$ when
$m = −15, n = 7$

Solution

$−8$

$p + q$ when
$p = −9, q = 17$

$r + s$ when $r = −9, s = −7$

Solution

$−16$

$t + u$ when $t = −6, u = −5$

${(x + y)}^{2}$ when
$x = −3, y = 14$

Solution

121

${(y + z)}^{2}$ when
$y = −3, z = 15$

$−2 x + 17$ when

ⓐ $x = 8$
ⓑ $x = −8$

Solution

ⓐ 1
ⓑ 33

$−5 y + 14$ when
ⓐ $y = 9$
ⓑ $y = −9$

$10 - 3 m$ when
ⓐ $m = 5$
ⓑ $m = −5$

Solution

ⓐ $−5$ ⓑ 25

$18 - 4 n$ when
ⓐ $n = 3$
ⓑ $n = −3$

$2 w^{2} - 3 w + 7$ when
$w = −2$

Solution

21

$3 u^{2} - 4 u + 5$ when $u = −3$

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

Solution

$−56$

$7 m - 4 n - 2$ 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

Solution

$(3 + (−15)) + 7; - 5$

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

the difference of 10 and $−18$

Solution

$10 - (−18); 28$

subtract 11 from $−25$

the difference of $−5$ and $−30$

Solution

$−5 - (−30); 25$

subtract $−6$ from $−13$

the product of $−3 and 15$

Solution

$−3 \cdot 15; - 45$

the product of $−4 and 16$

the quotient of $−60$ and $−20$

Solution

$−60 \div (−20); 3$

the quotient of $−40$ and $−20$

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

Solution

$\frac{−6}{a + b}$

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

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

Solution

$−10 (p - q)$

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?

Solution

$96 °$

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?

Solution

21

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?

Solution

$− $28$

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?

Solution

$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?

Solution

$− $3600$

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.

Solution

Answers may vary

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

Why is $− 2^{4} \neq {(−2)}^{4} ?$

Solution

Answers may vary

Why is $− 4^{3} = {(−4)}^{3} ?$

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?