Introduction

We can use the standard algorithm to divide three-digit decimals by two-digit decimals.

As an example, let's solve the following division problem:

92.4 \div 22

We start by writing down our problem using long division notation, keeping the decimal point in the dividend:

22 \!\require{enclose}\enclose{longdiv}{9 {\:\phantom{|}} 2 {\,\bbox[2px, lightgray]{\color{blue}.}\,} 4}

We also bring the decimal point up into the quotient, placing it directly above the decimal point in the dividend:

\: \phantom{0} {\:\phantom{|}} \phantom{0} {\:\phantom{|}} \bbox[2px, lightgray]{\color{blue}.}
22 \!\require{enclose}\enclose{longdiv}{9 {\:\phantom{|}} 2 {\,\bbox[2px, lightgray]{\color{blue}.}\,} 4}

Now, сarrying out the long division process, as usual, we get:

4 \!\!\! . 2
22 \!\require{enclose}\enclose{longdiv}{9 {\:\phantom{|}} 2 {\, . \,} 4}
-\!\!\!\! 8 8
4 4
-\!\!\!\! 4 4
0

Therefore, we conclude that 92.4 \div 22 = 4.2\,.

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Example: Dividing a Three-Digit Decimal Greater Than One by a Whole Number

What is 5.98 \div 13?

EXPLANATION

First, we write down the problem using long division notation, keeping the decimal point in the dividend:

13 \!\require{enclose}\enclose{longdiv}{5 {\,\bbox[2px, lightgray]{\color{blue}.}\,} 9 {\:\phantom{|}} 8}

We also bring the decimal point up into the quotient, placing it directly above the decimal point in the dividend:

\! \phantom{0} \bbox[2px, lightgray]{\color{blue}.}
13 \!\require{enclose}\enclose{longdiv}{5 {\,\bbox[2px, lightgray]{\color{blue}.}\,} 9 {\:\phantom{|}} 8}

Now, сarrying out the long division process, as usual, we get:

0 \!. 4 6
13 \!\require{enclose}\enclose{longdiv}{5 {\, . \,} 9 {\:\phantom{|}} 8}
-\!\!\!\! 5 2
7 8
-\!\!\!\! 7 8
0

Therefore, we conclude that 5.98 \div 13 = 0.46\,.

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Practice: Dividing a Three-Digit Decimal Greater Than One by a Whole Number

Calculate $3.92 \div 35.$

a
 $0.109$
b
 $0.110$
c
 $0.212$
d
 $0.111$
e
 $0.112$
Practice: Dividing a Three-Digit Decimal Greater Than One by a Whole Number

What is $27.6 \div 23?$

a
 $0.15$
b
 $1.4$
c
 $1.2$
d
 $1.3$
e
 $14$
Example: Dividing a Three-Digit Decimal Less Than One by a Whole Number

Calculate 0.24 \div 20.

EXPLANATION

First, we write down the problem using long division notation, keeping the decimal point in the dividend:

20 \!\require{enclose}\enclose{longdiv}{0 {\,\bbox[2px, lightgray]{\color{blue}.}\,} 2 {\:\phantom{|}} 4}

We also bring the decimal point up into the quotient, placing it directly above the decimal point in the dividend:

\! {\:\phantom{|}} \bbox[2px, lightgray]{\color{blue}.}
20\, \!\require{enclose}\enclose{longdiv}{0 {\,\bbox[2px, lightgray]{\color{blue}.}\,} 2 {\:\phantom{|}} 4}

Now, сarrying out the long division process, as usual, we get:

0 \!\! . 0 1 2
20\: \!\!\require{enclose}\enclose{longdiv}{ 0 {\, . \,} 2 {\:\phantom{|}} 4 {\:\phantom{|}} 0 }
-\!\!\!\! 2 0
4 0
-\!\!\!\! 4 0
0

Therefore, we conclude that 0.24 \div 20 = 0.012\,.

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Practice: Dividing a Three-Digit Decimal Less Than One by a Whole Number

Calculate $0.66 \div 30.$

a
 $0.024$
b
 $2.2$
c
 $0.022$
d
 $0.032$
e
 $0.22$
Practice: Dividing a Three-Digit Decimal Less Than One by a Whole Number

Calculate $0.36 \div 15.$

a
 $0.034$
b
 $2.4$
c
 $0.024$
d
 $0.028$
e
 $0.25$
Dividing Decimals by Decimals Using the Standard Algorithm

Using the standard algorithm requires the divisor to be a whole number. However, we can often manipulate a division problem using our understanding of fractions to give an equivalent problem where the standard algorithm can be applied.

As an example, let's consider the following division problem:

8.82 \div 2.5

We cannot apply the standard algorithm because the divisor (2.5) is not a whole number.

However, if we express this division problem as a fraction, we get

\dfrac{8.82}{2.5}.

Now, we need to create a whole number in the denominator. To do this, we make an equivalent fraction by multiplying both the numerator and denominator by 10\mathbin{:}

\dfrac{8.82 \times 10}{2.5 \times 10} = \dfrac{88.2}{25}

So, instead of our original problem, we can now solve the equivalent problem

88.2 \div 25.

Let's write down the problem using long division notation, keeping the decimal point in the dividend:

25 \!\require{enclose}\enclose{longdiv}{8 {\:\phantom{|}}8 {\,\bbox[2px, lightgray]{\color{blue}.}\,} 2 }

We also bring the decimal point up into the quotient, placing it directly above the decimal point in the dividend:


\:\: {\:\phantom{||}} {\:\phantom{||}} {\phantom{||}} \bbox[2px, lightgray]{\color{blue}.}
25 \!\require{enclose}\enclose{longdiv}{8 {\:\phantom{|}}8 {\,\bbox[2px, lightgray]{\color{blue}.}\,} 2 }

Now, сarrying out the long division process, as usual, we get:


3 \!\! . 5 2 8
\!25 \!\!\require{enclose}\enclose{longdiv}{\,8 {\:\phantom{|}} 8 {\, . \,}2 {\:\phantom{|}} 0 {\:\phantom{|}} 0}
-\!\!\!\! 7 5
1 3 2
-\!\!\!\! 1 2 5
7 0
-\!\!\!\! 5 0
2 0 0
-\!\!\!\! 2 0 0
0

Therefore, we conclude that 8.82 \div 2.5 = 3.528 \,.

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Example: Dividing Decimals by Decimals

Dave was paid \[math]7.47 for 1.2 hours of work. What was Dave's hourly rate?

EXPLANATION

To determine Dave's hourly rate, we have to calculate

7.47 \div 1.2.

We can write this division problem as a fraction, as follows:

\dfrac{7.47}{1.2}

Multiplying the numerator and denominator by 10, we get a whole number in the denominator:

\dfrac{7.47 \times 10}{1.2 \times 10} = \dfrac{74.7}{12}

So the problem is equivalent to 74.7 \div 12.

Next, we write down the problem using long division notation, keeping the decimal point in the dividend:

12 \!\require{enclose}\enclose{longdiv}{7 {\:\phantom{|}} 4 {\,\bbox[2px, lightgray]{\color{blue}.}\,} 7 }

We also bring the decimal point up into the quotient, placing it directly above the decimal point in the dividend:

\:\:\: \phantom{0} {\:\phantom{|}} \phantom{0} {\:\phantom{|}} \bbox[2px, lightgray]{\color{blue}.}
12 \!\require{enclose}\enclose{longdiv}{7 {\:\phantom{|}} 4 {\,\bbox[2px, lightgray]{\color{blue}.}\,} 7 }

Now, сarrying out the long division process, as usual, we get:

6 \!\! . 2 2 5
12 \!\!\require{enclose}\enclose{longdiv}{7 {\:\phantom{|}} 4 {\, . \,} 7 {\:\phantom{|}} 0 {\:\phantom{|}} 0}
-\!\!\!\! 7 2
2 7
-\!\!\! 2 4
3 0
-\!\!\! 2 4
6 0
-\!\!\! 6 0
0

So, 7.47 \div 1.2 = 6.225.

Therefore, Dave was paid \[math]6.225 per hour.

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Practice: Dividing Decimals by Decimals

A plane covered a distance of $6.57$ miles in $1.5$ minutes, moving at a constant rate. What distance did the plane cover in one minute?

a
 $4.07$ miles
b
 $4.38$ miles
c
 $3.86$ miles
d
 $3.36$ miles
e
 $2.78$ miles
Practice: Dividing Decimals by Decimals

Find the value of $1.24 \div 1.6.$

a
 $0.765$
b
 $0.657$
c
 $0.665$
d
 $0.775$
e
 $0.675$
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