# 10.6: Multiplication and Division of Signed Numbers

- Page ID
- 48897

Learning Objectives

- be able to multiply and divide signed numbers
- be able to multiply and divide signed numbers using a calculator

## Multiplication of Signed Numbers

Let us consider first, the product of two positive numbers. Multiply: \(3 \cdot 5\).

\(3 \cdot 5\) means \(5 + 5 + 5 = 15\)

This suggests that (In later mathematics courses, the word "suggests" turns into the word "proof." One example does not prove a claim. Mathematical proofs are constructed to validate a claim for all possible cases.)

\(\text{(positive number)} \cdot \text{(positive number)} = \text{(positive number)}\)

More briefly,

(+) (+) = (+)

Now consider the product of a positive number and a negative number. Multiply: (3)(-5)

(3)(-5) means (-5) + (-5) + (-5) = -15

This suggests that

\(\text{(positive number)} \cdot \text{(negative number)} = \text{(negative number)}\)

More briefly,

(+) (-) = (-)

By the commutative property of multiplication, we get

\(\text{(negative number)} \cdot \text{(positive number)} = \text{(negative number)}\)

More briefly,

(-) (+) = (-)

The sign of the product of two negative numbers can be suggested after observing the following illustration.

Multiply -2 by, respectively, 4, 3, 2, 1, 0, -1, -2, -3, -4.

We have the following rules for multiplying signed numbers.

**Rules for Multiplying Signed ****Numbers**

Multiplying signed numbers:

- To multiply two real numbers that have the
*same sign*, multiply their absolute values. The product is positive.

(+) (+) = (+)

(-) (-) = (+) - To multiply two real numbers that have
*opposite signs*, multiply their absolute values. The product is negative.

(+) (-) = (-)

(-) (+) = (-)

Sample Set A

Find the following products.

\(8 \cdot 6\)

**Solution**

\(\begin{array} {ccl} {|8|} & = & {8} \\ {|6|} & = & {6} \end{array} \big \}\) Multiply these absolute values.

\(8 \cdot 6 = 48\)

Since the numbers have the same sign, the product is positive.

Thus, \(8 \cdot 6 = +48\), or \(8 \cdot 6 = 48\).

Sample Set A

(-8)(-6)

**Solution**

\(\begin{array} {ccl} {|-8|} & = & {8} \\ {|-6|} & = & {6} \end{array} \big \}\) Multiply these absolute values.

\(8 \cdot 6 = 48\)

Since the numbers have the same sign, the product is positive.

Thus, \((-8)(-6) = +48\), or \((-8)(-6) = 48\).

Sample Set A

(-4)(7)

**Solution**

\(\begin{array} {ccl} {|-4|} & = & {4} \\ {|7|} & = & {7} \end{array} \big \}\) Multiply these absolute values.

\(4 \cdot 7 = 28\)

Since the numbers have opposite signs, the product is negative.

Thus, (-4)(7) = -28.

Sample Set A

6(-3)

**Solution**

\(\begin{array} {ccl} {|6|} & = & {6} \\ {|-3|} & = & {3} \end{array} \big \}\) Multiply these absolute values.

\(6 \cdot 3 = 18\)

Since the numbers have opposite signs, the product is negative.

Thus, 6(-3) = -18.

Practice Set A

Find the following products.

3(-8)

**Answer**-
-24

Practice Set A

4(16)

**Answer**-
64

Practice Set A

(-6)(-5)

**Answer**-
30

Practice Set A

(-7)(-2)

**Answer**-
14

Practice Set A

(-1)(4)

**Answer**-
-4

Practice Set A

(-7)7

**Answer**-
-49

## Division of Signed Numbers

To determine the signs in a division problem, recall that

\(\dfrac{12}{3} = 4\) since \(12 = 3 \cdot 4\)

This suggests that

\(\dfrac{(+)}{(+)} = (+)\)

\(\dfrac{(+)}{(+)} = (+)\) since (+) = (+) (+)

What is \(\dfrac{12}{-3}\)?

12 = (-3)(-4) suggets that \(\dfrac{12}{-3} = -4\). That is,

\(\dfrac{(+)}{(-)} = (-)\)

(+) = (-) (-) suggets that \(\dfrac{(+)}{(-)} = (-)\)

What is \(\dfrac{-12}{3}\)?

-12 = (3)(-4) suggests that \(\dfrac{-12}{3} = -4\). That is,

\(\dfrac{(-)}{(+)} = (-)\)

(-) = (+) (-) suggets that \(\dfrac{(-)}{(+)} = (-)\)

What is \(\dfrac{-12}{-3}\)?

-12 = (-3)(4) suggets that \(\dfrac{-12}{-3} = 4\). That is,

\(\dfrac{(-)}{(-)} = (+)\)

(-) = (-)(+) suggests that \(\dfrac{(-)}{(-)} = (+)\)

We have the following rules for dividing signed numbers.

**Rules for Dividing Signed Numbers**

Dividing signed numbers:

- To divide two real numbers that have the
*same sign*, divide their absolute values. The quotient is positive.

\(\dfrac{(+)}{(+)} = (+)\dfrac{(-)}{(-)} = (+)\) - To divide two real numbers that have
*opposite signs*, divide their absolute values. The quotient is negative.

\(\dfrac{(-)}{(+)} = (-)\dfrac{(+)}{(-)} = (-)\)

Sample Set B

Find the following quotients.

\(\dfrac{-10}{2}\)

**Solution**

\(\begin{array} {ccc} {|-10|} & = & {10} \\ {|2|} & = & {2} \end{array} \big \}\) Divide these absolute values.

\(\dfrac{10}{2} = 5\)

Since the numbers have opposite signs, the quotient is negative.

Thus \(\dfrac{-10}{2} = -5\).

Sample Set B

\(\dfrac{-35}{-7}\)

**Solution**

\(\begin{array} {ccc} {|-35|} & = & {35} \\ {|-7|} & = & {7} \end{array} \big \}\) Divide these absolute values.

\(\dfrac{35}{7} = 5\)

Since the numbers have the same signs, the quotient is positive.

Thus \(\dfrac{-35}{-7} = 5\).

Sample Set B

\(\dfrac{18}{-9}\)

**Solution**

\(\begin{array} {ccc} {|18|} & = & {18} \\ {|-9|} & = & {9} \end{array} \big \}\) Divide these absolute values.

\(\dfrac{18}{9} = 2\)

Since the numbers have opposite signs, the quotient is negative.

Thus \(\dfrac{18}{-9} = -2\).

Practice Set B

Find the following quotients.

\(\dfrac{-24}{-6}\)

**Answer**-
4

Practice Set B

\(\dfrac{30}{-5}\)

**Answer**-
-6

Practice Set B

\(\dfrac{-54}{27}\)

**Answer**-
-2

Practice Set B

\(\dfrac{51}{17}\)

**Answer**-
3

Sample Set C

Find the value of \(\dfrac{-6(4 - 7) - 2(8 - 9)}{-(4 + 1) + 1}\).

**Solution**

Using the order of operations and what we know about signed numbers, we get,

\(\begin{array} {rcl} {\dfrac{-6(4 - 7) - 2(8 - 9)}{-(4 + 1) + 1}} & = & {\dfrac{-6(-3) - 2(-1)}{-(5) + 1}} \\ {} & = & {\dfrac{18 + 2}{-5 + 1}} \\ {} & = & {\dfrac{20}{-4}} \\ {} & = & {-5} \end{array}\)

Practice Set C

Find the value of \(\dfrac{-5(2 - 6) - 4(-8 - 1)}{2(3 - 10) - 9(-2)}\).

**Answer**-
14

## Calculators

Calculators with the

key can be used for multiplying and dividing signed numbers.

Sample Set D

Use a calculator to find each quotient or product.

\((-186) \cdot (-43)\)

**Solution**

Since this product involves a \(\text{(negative)} \cdot \text{(negative)}\), we know the result should be a positive number. We'll illustrate this on the calculator.

Display Reads | ||

Type | 186 | 186 |

Press | -186 | |

Press | \(\times\) | -186 |

Type | 43 | 43 |

Press | -43 | |

Press | = | 7998 |

Thus, \((-186) \cdot (-43) = 7,998\)

Sample Set D

\(\dfrac{158.64}{-54.3}\). Round to one decimal place.

**Solution**

Since this product involves a \(\text{(negative)} \cdot \text{(negative)}\), we know the result should be a positive number. We'll illustrate this on the calculator.

Display Reads | ||

Type | 158.64 | 158.64 |

Press | \(\div\) | 158.64 |

Type | 54.3 | 54.3 |

Press | -54.3 | |

Press | = | -2.921546961 |

Rounding to one decimal place we get -2.9.

Practice Set D

Use a calculator to find each value.

\((-51.3) \cdot (-21.6)\)

**Answer**-
1,108.08

Practice Set D

\(-2.5746 \div -2.1\)

**Answer**-
1.226

Practice Set D

\((0.006) \cdot (-0.241)\). Round to three decimal places.

**Answer**-
-0.001

## Exercises

Find the value of each of the following. Use a calculator to check each result.

Exercise \(\PageIndex{1}\)

(-2)(-8)

**Answer**-
16

Exercise \(\PageIndex{2}\)

(-3)(-9)

Exercise \(\PageIndex{3}\)

(-4)(-8)

**Answer**-
32

Exercise \(\PageIndex{4}\)

(-5)(-2)

Exercise \(\PageIndex{5}\)

(3)(-12)

**Answer**-
-36

Exercise \(\PageIndex{6}\)

(4)(-18)

Exercise \(\PageIndex{7}\)

(10)(-6)

**Answer**-
-60

Exercise \(\PageIndex{8}\)

(-6)(4)

Exercise \(\PageIndex{9}\)

(-2)(6)

**Answer**-
-12

Exercise \(\PageIndex{10}\)

(-8)(7)

Exercise \(\PageIndex{11}\)

\(\dfrac{21}{7}\)

**Answer**-
3

Exercise \(\PageIndex{12}\)

\(\dfrac{42}{6}\)

Exercise \(\PageIndex{13}\)

\(\dfrac{-39}{3}\)

**Answer**-
-13

Exercise \(\PageIndex{14}\)

\(\dfrac{-20}{10}\)

Exercise \(\PageIndex{15}\)

\(\dfrac{-45}{-5}\)

**Answer**-
9

Exercise \(\PageIndex{16}\)

\(\dfrac{-16}{-8}\)

Exercise \(\PageIndex{17}\)

\(\dfrac{25}{-5}\)

**Answer**-
-5

Exercise \(\PageIndex{18}\)

\(\dfrac{36}{-4}\)

Exercise \(\PageIndex{19}\)

8 - (-3)

**Answer**-
11

Exercise \(\PageIndex{20}\)

14 - (-20)

Exercise \(\PageIndex{21}\)

20 - (-8)

**Answer**-
28

Exercise \(\PageIndex{22}\)

(-4) - (-1)

Exercise \(\PageIndex{23}\)

0 - 4

**Answer**-
-4

Exercise \(\PageIndex{24}\)

0 - (-1)

Exercise \(\PageIndex{25}\)

-6 + 1 - 7

**Answer**-
-12

Exercise \(\PageIndex{26}\)

15 - 12 - 20

Exercise \(\PageIndex{27}\)

1 - 6 - 7 + 8

**Answer**-
-4

Exercise \(\PageIndex{28}\)

2 + 7 - 10 + 2

Exercise \(\PageIndex{29}\)

3(4 - 6)

**Answer**-
-6

Exercise \(\PageIndex{30}\)

8(5 - 12)

Exercise \(\PageIndex{31}\)

-3(1 - 6)

**Answer**-
15

Exercise \(\PageIndex{32}\)

-8(4 - 12) + 2

Exercise \(\PageIndex{33}\)

-4(1 - 8) + 3(10 - 3)

**Answer**-
49

Exercise \(\PageIndex{34}\)

-9(0 - 2) + 4(8 - 9) + 0(-3)

Exercise \(\PageIndex{35}\)

6(-2 - 9) - 6(2 + 9) + 4(-1 - 1)

**Answer**-
-140

Exercise \(\PageIndex{36}\)

\(\dfrac{3(4 + 1) - 2 (5)}{-2}\)

Exercise \(\PageIndex{37}\)

\(\dfrac{4(8 + 1) - 3 (-2)}{-4 - 2}\)

**Answer**-
-7

Exercise \(\PageIndex{38}\)

\(\dfrac{-1(3 + 2) + 5}{-1}\)

Exercise \(\PageIndex{39}\)

\(\dfrac{-3(4 - 2) + (-3)(-6)}{-4}\)

**Answer**-
-3

Exercise \(\PageIndex{40}\)

-1(4 + 2)

Exercise \(\PageIndex{41}\)

-1(6 - 1)

**Answer**-
-5

Exercise \(\PageIndex{42}\)

-(8 + 21)

Exercise \(\PageIndex{43}\)

-(8 - 21)

**Answer**-
13

#### Exercises for Review

Exercise \(\PageIndex{44}\)

Use the order of operations to simplify \((5^2 + 3^2 + 2) \div 2^2\).

Exercise \(\PageIndex{45}\)

Find \(\dfrac{3}{8}\) of \(\dfrac{32}{9}\).

**Answer**-
\(\dfrac{4}{3} = 1 \dfrac{1}{3}\)

Exercise \(\PageIndex{46}\)

Write this number in decimal form using digits: “fifty-two three-thousandths”

Exercise \(\PageIndex{47}\)

The ratio of chlorine to water in a solution is 2 to 7. How many mL of water are in a solution that contains 15 mL of chlorine?

**Answer**-
\(52 \dfrac{1}{2}\)

Exercise \(\PageIndex{48}\)

Perform the subtraction -8 - (-20)