Introduction

Dividing with negative numbers follows similar rules as multiplying with negative numbers.

First of all, a negative number divided by a positive number gives a negative number. We can picture this rule as \begin{align*} \mathbf{\color{red}(-)} \:\mathbf{\div}\: \mathbf{\color{blue}(+)} \:=\: \mathbf{\color{red}(-)} . \end{align*}

For example, let's use this rule to find the value of (-6) \div 2.

Here, since we are dividing a negative number \color{red}(-6) by a positive number {\color{blue}(2)}, the result is going to be negative: {\color{red}(-6)} \div {\color{blue}2} = {\color{red} \mathbf{-} \fbox{[math]\phantom{3}[/math]} }

Now, to compute the number that goes in the box, we divide 6 by 2 without any signs: 6 \div 2 = 3. So, we get {\color{red}(-6)} \div {\color{blue}2} = {\color{red} -3 }.

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Example: Dividing a Negative Number by a Positive Number

Find the value of -15 \div 3.

EXPLANATION

Here, we are dividing a negative number \color{red}(-15) by a positive one \color{blue}(3). So the result will be negative: {\color{red}(-15)} \div {\color{blue}3} = {\color{red}\mathbf{-} \fbox{[math]\phantom{5}[/math]} }

Now, to compute the number that goes in the box, we divide 15 by 3 without any signs: 15 \div 3 = 5. Therefore, we get {\color{red}(-15)} \div {\color{blue}3} = {\color{red}-5}.

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Practice: Dividing a Negative Number by a Positive Number

$(-18) \div 6=$

a
 $-12$
b
 $4$
c
 $-4$
d
 $3$
e
 $-3$
Practice: Dividing a Negative Number by a Positive Number

Expressed as a fraction, $(-7) \div 3=$

a
 $\dfrac{1}{3}$
b
 $-\dfrac{7}{3}$
c
 $-\dfrac{1}{3}$
d
 $-\dfrac{3}{7}$
e
 $\dfrac{7}{3}$
Practice: Dividing a Negative Number by a Positive Number

Expressed as a mixed number, $(-25) \div 6=$

a
 $-4\,\dfrac{1}{6}$
b
 $-4\,\dfrac{5}{6}$
c
 $3\,\dfrac{5}{6}$
d
 $4\,\dfrac{1}{6}$
e
 $-3\,\dfrac{5}{6}$
The Quotient of a Positive and a Negative is a Negative

A positive number divided by a negative number is also a negative number. We can picture this rule as \begin{align*} \mathbf{\color{blue}(+)} \:\mathbf{\div}\: \mathbf{\color{red}(-)} \:=\: \mathbf{\color{red}(-)} . \end{align*}

For instance, to find 8\div (-2), we have to divide a positive number \color{blue}(8) by a negative one {\color{red}(-2)}. So the result will be negative: {\color{blue}8} \div {\color{red}(-2)} = {\color{red}\mathbf{-} \fbox{[math]\phantom{10}[/math]} }

Now, to compute the number that goes in the box, we divide 8 by 2 without any signs: 8 \div 2 = 4. So, we get {\color{blue}8} \div {\color{red}(-2)} = {\color{red}-4}.

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Example: Dividing a Positive Number by a Negative Number

Calculate the value of 50 \div (-5).

EXPLANATION

We have to divide a positive number \color{blue}(50) by a negative one \color{red}(-5). So the result will be negative: {\color{blue}{50}} \div {\color{red}{(-5)}} = {\color{red}{\mathbf{-} \fbox{[math]\phantom{10}[/math]} }}

Now, to compute the number that goes in the box, we divide 50 by 5 without any signs: 50 \div 5 = 10. Therefore, we get {\color{blue}{50}} \div {\color{red}{(-5)}} = {\color{red}{-10}}.

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Practice: Dividing a Positive Number by a Negative Number

$9 \div(-3)=$

a
 $-27$
b
 $-3$
c
 $3$
d
 $27$
e
 $-12$
Practice: Dividing a Positive Number by a Negative Number

Expressed as a mixed number, $19 \div (-6)=$

a
 $-3\,\dfrac{1}{6}$
b
 $2\,\dfrac{5}{6}$
c
 $-3\,\dfrac{5}{6}$
d
 $3\,\dfrac{1}{6}$
e
 $-2\,\dfrac{5}{6}$
The Quotient of a Negative and a Negative is a Positive

Finally, the third rule is that a negative number divided by a negative number is a positive number:

\begin{align*} \mathbf{\color{red}(-)} \:\mathbf{\div}\: \mathbf{\color{red}(-)} \:=\: \mathbf{\color{blue}(+)} \end{align*}

To demonstrate, let's find -12 \div (-3). We have to divide a negative number \color{red}(-12) by another negative number {\color{red}(-3)}. So the result will be positive: {\color{red}(-12)} \div {\color{red}(-3)} = {\color{blue} \fbox{[math]\phantom{4}[/math]} }

Now, to compute the number that goes in the box, we divide 12 by 3 without any signs: 12 \div 3 = 4. Therefore, we get {\color{red}(-12)} \div {\color{red}(-3)} = {\color{blue}4}.

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Example: Dividing a Negative Number by a Negative Number

Calculate -81\div (-9).

EXPLANATION

We have to divide a negative number {\color{red}(-81)} by another negative number {\color{red}(-9)}. So the result will be positive: {\color{red}(-81)} \div {\color{red}(-9)} = {\color{blue} \fbox{[math]\phantom{9}[/math]} }

Now, to compute the number that goes in the box, we divide 81 by 9 without any signs: 81 \div 9 = 9. Therefore, we get {\color{red}(-81)} \div {\color{red}(-9)} = {\color{blue}9}.

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Practice: Dividing a Negative Number by a Negative Number

$(-1)\div(-1)=$

a
 $2$
b
 $-1$
c
 $-2$
d
 $0$
e
 $1$
Practice: Dividing a Negative Number by a Negative Number

Expressed as a fraction, $(-9) \div (-4)=$

a
 $\dfrac{4}{9}$
b
 $\dfrac{9}{4}$
c
 $-\dfrac{1}{4}$
d
 $-\dfrac{4}{9}$
e
 $-\dfrac{9}{4}$
Practice: Dividing a Negative Number by a Negative Number

Expressed as a mixed number, $(-15) \div (-4)=$

a
 $2\,\dfrac{3}{4}$
b
 $-3\,\dfrac{3}{4}$
c
 $-4\,\dfrac{1}{4}$
d
 $3\,\dfrac{3}{4}$
e
 $4\,\dfrac{1}{4}$
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