Introduction

There are many ways to divide decimals. One such way is to write the division as a fraction and then multiply the numerator and denominator by a power of 10 so that both become whole numbers. Then, we carry out the division as usual.

To illustrate, let's use this method to find the value of 4.5\div 0.5.

Step 1. First, we write the division problem as a fraction: \dfrac{4.5}{0.5}

Step 2. Then, we multiply the numerator and denominator by 10\mathbin{:} \dfrac{4.5\times 10}{0.5\times 10} = \dfrac{45}{5}

Step 3. Finally, we solve the resulting division problem. In this case, we have 45 \div 5 = 9.

Therefore, 4.5 \div 0.5 = 9 \, .

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Example: Dividing a Two-Digit Decimal by a Two-Digit Decimal

What is 7.2 divided by 0.6?

EXPLANATION

First, we write the division problem as a fraction:

\dfrac{7.2}{0.6}

Then, we multiply the numerator and denominator by 10 so that both become whole numbers:

\dfrac{7.2\times 10}{0.6\times 10} = \dfrac{72}{6}

So the problem is equivalent to 72\div 6. We will use long division:

6 \!\!\require{enclose}\enclose{longdiv}{7 {\:\phantom{|}} 2}

First, we consider the tens:

\color{blue}1
6 \!\!\require{enclose}\enclose{longdiv}{{\color{red}7} {\:\phantom{|}} 2}
-\!\!\!\! 6 \color{lightgray}\downarrow
1 \color{lightgray}2

Divide: {\color{red}7} \div 6 = {\color{blue}1}\,\text{R}\,1
Write over tens: \color{blue}1
Multiply: 6 \times {\color{blue}1} = 6
Subtract: {\color{red}7} - 6 = 1
Bring down: 2

Next, we consider the ones:

1 \color{blue}2
6 \!\!\require{enclose}\enclose{longdiv}{7 {\:\phantom{|}} 2}
-\!\!\!\! 6
\color{red}1 \color{red}2
-\!\!\!\! 1 2
\fbox{0}

Divide: {\color{red}12} \div 6 = {\color{blue}2}
Write over ones: \color{blue}2
Multiply: 6 \times {\color{blue}2} = 12
Subtract: {\color{red}12} - 12 = \fbox{0}

We've gone through all the digits, so the division is done.

We find that 72 \div 6 = 12, so we conclude that

7.2 \div 0.6 = 12 \, .

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Practice: Dividing a Two-Digit Decimal by a Two-Digit Decimal

What is $9.6$ divided by $0.4?$

a
 $26$
b
 $32$
c
 $22$
d
 $24$
e
 $18$
Practice: Dividing a Two-Digit Decimal by a Two-Digit Decimal

What is $1.2$ divided by $0.2?$

a
 $8$
b
 $6$
c
 $0.6$
d
 $4$
e
 $60$
Example: Dividing a Three-Digit Decimal by a Two-Digit Decimal

What is 86.5 \div 0.5?

EXPLANATION

First, we write the division problem as a fraction:

\dfrac{86.5}{0.5}

Then, we multiply the numerator and denominator by 10 so that both become whole numbers:

\dfrac{86.5\times 10}{0.5\times 10} = \dfrac{865}{5}

So the problem is equivalent to 865\div 5. We will use long division:

5 \!\!\require{enclose}\enclose{longdiv}{8 {\:\phantom{|}} 6 {\:\phantom{|}} 5}

Carrying out the long division process as usual, we get:

1 7 3
5 \!\!\require{enclose}\enclose{longdiv}{8 {\:\phantom{|}} 6 {\:\phantom{|}} 5}
-\!\!\!\!\! 5
3 6
-\!\!\!\!\! 3 5
1 5
-\!\!\!\!\! 1 5
{0}

We've gone through all the digits, so the division is done.

We find that 865 \div 5 = 173, so we conclude that

86.5 \div 0.5 = 173 \, .

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Practice: Dividing a Three-Digit Decimal by a Two-Digit Decimal

What is $56.4\div 0.4?$

a
 $143$
b
 $147$
c
 $141$
d
 $151$
e
 $149$
Practice: Dividing a Three-Digit Decimal by a Two-Digit Decimal

What is $62.4\div 0.8?$

a
 $73$
b
 $80.5$
c
 $78$
d
 $76.5$
e
 $81$
Example: Dividing a Three-Digit Decimal by a Three-Digit Decimal

What is 9.45 divided by 0.09?

EXPLANATION

First, we write the division problem as a fraction:

\dfrac{9.45}{0.09}

Then, we multiply the numerator and denominator by a power of 10 so that both become whole numbers. Here, there are 2 decimal places, so we need to multiply by 100\mathbin{:}

\dfrac{9.45\times 100}{0.09\times 100} = \dfrac{945}{9}

So the problem is equivalent to 945\div 9. We will use long division:

9 \!\!\require{enclose}\enclose{longdiv}{9 {\:\phantom{|}} 4 {\:\phantom{|}} 5}

Applying the long division process as usual, we get:

1 0 5
9 \!\!\require{enclose}\enclose{longdiv}{9 {\:\phantom{|}} 4 {\:\phantom{|}} 5}
-\!\!\!\!\! 9
0 4 5
-\!\!\!\! 4 5
0

We've gone through all the digits, so the division is done.

We found that 945 \div 9 = 105, so we conclude that 9.45 \div 0.09 = 105 \, .

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Practice: Dividing a Three-Digit Decimal by a Three-Digit Decimal

What is $7.63$ divided by $0.07 ?$

a
 $111.5$
b
 $109.5$
c
 $107$
d
 $109$
e
 $111$
Practice: Dividing a Three-Digit Decimal by a Three-Digit Decimal

What is $8.16$ divided by $0.08 ?$

a
 $102.2$
b
 $103.7$
c
 $107$
d
 $102$
e
 $101.7$
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