Along with multiplying decimal numbers, division of decimal numbers is another key operation that’s important to become comfortable with.
Dividing by 10, 100, 1000, …
When we wish to divide decimals by whole numbers, the simplest case is by a power of 10.Such as 10 or 100, the process for division is quite straight forward.
How large the power of 10 being used to divide is, is what determines how many places we move the decimal point to the left in the answer.
Divide by 10 …… Move decimal point 1 place left.
Divide by 100 …… Move decimal point 2 places left.
Divide by 1000 …… Move decimal point 3 places left. etc
Examples
1.1
a) 2.4 ÷ 10 = 0.24 b) 2.4 ÷ 100 = 0.024
1.2
0.4 ÷ 100 = 0.004
1.3
0.5 ÷ 1000 = 0.0005
1.4
7.3 ÷ 10’000 = 0.00073
Divide Decimals by Whole Numbers
Other Whole Numbers
An effective approach for when we wish to divide decimals by whole numbers with sums such as, 6.48 ÷ 2.
Is to address it with long division as if it was 648 ÷ 2, you just have to pay attention to where the decimal point is placed in the sum.
The decimal point in the quotient answer part is placed directly above where it appears in the dividend.
This can be best observed in some examples.
Examples
2.1
4.12 ÷ 4
Solution
First we set up as a normal division sum as standard.
\begin{array}{r} \\[-2pt] 4 {\bf{|}} {\overline{\space 4.12 \space\space}} \end{array}Then we place a decimal point in the place where the answer will be, directly above the place of the decimal point in the dividend below.
\begin{array}{r} . \space\space\space\space\space\space \\[-2pt] 4 {\bf{|}} {\overline{\space 4.12 \space\space}} \end{array}
Now from this stage it’s standard long division steps.
\begin{array}{r} 1. \space\space\space\space\space\space\space \\[-2pt] 4 {\bf{|}} {\overline{\space 4.12\space}}\space\space \\ {\text{--}}\space \underline{ 4 \space\space\space\space}\space\space\space\space \\ 0\space\space \space\space\space\space\space\space \\ \space\space\space\space\space\space\space \\ \space\space\space\space\space\space\space\\ \space \\ \space \end{array} => \begin{array}{r} 1.0 \space\space\space\space\space \\[-2pt] 4 {\bf{|}} {\overline{\space 4.12\space}}\space\space \\ {\text{--}}\space \underline{ 4 \space\space\space\space}\space\space\space\space \\ 0\space 1\space\space\space\space\space \\ {\text{--}}\space \underline{\space\space0} \space\space\space\space\space \\ 1 \space\space\space\space\space\\ \space\\ \space \end{array} => \begin{array}{r} 1.03 \space\space\space \\[-2pt] 4 {\bf{|}} {\overline{\space 4.12\space}}\space\space \\ {\text{--}}\space \underline{ 4 \space\space\space\space}\space\space\space\space \\ 0\space1 \space\space\space\space\space \\ {\text{--}}\space \underline{\space\space0} \space\space\space\space\space \\ 12 \space\space\space\\ {\text{--}}\space \underline{12} \space\space\space \\ 0 \space\space\space \end{array}
4.12 ÷ 4 = 1.03
2.2
13.04 ÷ 4
Solution
\begin{array}{r} \\[-2pt] 4 {\bf{|}} {\overline{\space 13.04 \space\space}} \end{array}
\begin{array}{r} . \space\space\space\space\space\space \\[-2pt] 4 {\bf{|}} {\overline{\space 13.04 \space\space}} \end{array}
\begin{array}{r} 0\space\space. \space\space\space\space\space\space \\[-2pt] 4 {\bf{|}} {\overline{\space 13.04 \space\space}}\\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space \\ 1\space\space \space\space\space\space\space\space\space \\ \space\space\space\space\space\space\space \\ \space\space\space\space\space\space\space\\ \space \\ \space\\ \\ \\ \end{array} => \begin{array}{r} 03. \space\space\space\space\space\space \\[-2pt] 4 {\bf{|}} {\overline{\space 13.04 \space\space}}\\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space \\ 13 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{12} \space\space\space\space\space\space\space \\ 1 \space\space\space\space\space\space\space\\ \space\\ \space \\ \\ \\ \end{array} => \begin{array}{r} 03.2 \space\space\space\space \\[-2pt] 4 {\bf{|}} {\overline{\space 13.04 \space\space}}\\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space \\ 13 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{12} \space\space\space\space\space\space\space \\ 1\space0 \space\space\space\space\\ {\text{--}}\space \underline{8} \space\space\space\space \\ 2 \space\space\space\space\\ \\ \\ \end{array} => \begin{array}{r} 03.26 \space\space \\[-2pt] 4 {\bf{|}} {\overline{\space 13.04 \space\space}}\\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space \\ 13 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{12} \space\space\space\space\space\space\space \\ 1\space0 \space\space\space\space\\ {\text{--}}\space \underline{8} \space\space\space\space \\ 24 \space\space\\ {\text{--}}\space \underline{24} \space\space \\ 0 \space\space \\ \end{array}
13.04 ÷ 4 = 3.26
2.3
8.26 ÷ 5
Solution
\begin{array}{r} \\[-2pt] 5 {\bf{|}} {\overline{\space 8.26 \space\space}} \end{array}
\begin{array}{r} . \space\space\space\space\space\space \\[-2pt] 5 {\bf{|}} {\overline{\space 8.26 \space\space}} \end{array}
\begin{array}{r} 1. \space\space\space\space\space\space \\[-2pt] 5 {\bf{|}} {\overline{\space 8.26 \space\space}}\\ {\text{--}}\space \underline{ 5 \space\space\space}\space\space\space\space \\ 3\space\space \space\space\space\space\space \\ \space\space\space\space\space\space\space \\ \space\space\space\space\space\space\space\\ \space \\ \space\\ \\ \\ \end{array} => \begin{array}{r} 1.6 \space\space\space\space \\[-2pt] 5 {\bf{|}} {\overline{\space 8.26 \space\space}}\\ {\text{--}}\space \underline{ 5 \space\space\space}\space\space\space\space \\ 3\space2 \space\space\space\space \\ {\text{--}}\space \underline{3\space0} \space\space\space\space \\ 2 \space\space\space\space\\ \space\\ \space \\ \\ \\ \end{array} => \begin{array}{r} 1.65 \space\space \\[-2pt] 5 {\bf{|}} {\overline{\space 8.26 \space\space}}\\ {\text{--}}\space \underline{ 5 \space\space\space}\space\space\space\space \\ 3\space2 \space\space\space\space \\ {\text{--}}\space \underline{3\space0} \space\space\space\space \\ 26 \space\space\\ {\text{--}}\space \underline{25} \space\space \\ 1 \space\space\\ \\ \\ \end{array}
Now at this point one might think that the answer is 1.65 with remainder 1.
But this is not the case as we are dealing with decimals and not whole numbers.
We need to find a way to bring a digit down that doesn’t change the value of any numbers in the overall division sum.
We can do this by changing the divisor 8.26 to 8.260.
This is fine as 8.26 = 8.260.
\begin{array}{r} 1.652 \\[-2pt] 5 {\bf{|}} {\overline{\space 8.260 }}\\ {\text{--}}\space \underline{ 5 \space\space\space}\space\space\space\space \\ 3\space2 \space\space\space\space \\ {\text{--}}\space \underline{3\space0} \space\space\space\space \\ 26 \space\space\\ {\text{--}}\space \underline{25} \space\space \\ 10 \\ {\text{--}}\space \underline{10} \\ 0 \\ \end{array}
8.26 ÷ 5 = 1.652
Answers with no End or Repeating
The answers to examples (2.2) and (2.3) above had decimal answers that terminated and came to an end.
There are times when this isn’t the case however when we divide decimals by whole numbers.
The division of a decimal can sometimes result in another decimal that has no end and possibly even repeating.
We’ll look at such an example here.
3.1
8.249 ÷ 3
Solution
\begin{array}{r} 2.749 \space\space\space\space\space\space \\[-2pt] 3 {\bf{|}} {\overline{\space 8.249 \space\space\space\space\space\space}}\\ {\text{--}}\space \underline{ 6 \space\space}\space\space\space\space\space\space\space\space\space\space\space \\ 2\space2 \space\space\space\space\space\space\space\space\space\space \\ {\text{--}}\space \underline{2\space1} \space\space\space\space\space\space\space\space\space\space \\ 14 \space\space\space\space\space\space\space\space\\ {\text{--}}\space \underline{12} \space\space\space\space\space\space\space\space \\ 2 \space\space\space\space\space\space\space\space\\ 29 \space\space\space\space\space\space\\ {\text{--}}\space \underline{27} \space\space\space\space\space\space \\ 2 \space\space\space\space\space\space\\ \end{array}
We’ll now add an extra two 0‘s to the divisor and see if there is a pattern.
\begin{array}{r} 2.74966 \space\space \\[-2pt] 3 {\bf{|}} {\overline{\space 8.24900 \space\space}}\\ {\text{--}}\space \underline{ 6 \space\space}\space\space\space\space\space\space\space\space\space\space\space \\ 2\space2 \space\space\space\space\space\space\space\space\space\space \\ {\text{--}}\space \underline{2\space1} \space\space\space\space\space\space\space\space\space\space \\ 14 \space\space\space\space\space\space\space\space\\ {\text{--}}\space \underline{12} \space\space\space\space\space\space\space\space \\ 2 \space\space\space\space\space\space\space\space\\ 29 \space\space\space\space\space\space\\ {\text{--}}\space \underline{27} \space\space\space\space\space\space \\ 20 \space\space\space\space\\ {\text{--}}\space \underline{18} \space\space\space\space \\ 20 \space\space\space\space\\ {\text{--}}\space \underline{18} \space\space\space\space \\ 2 \space\space\space\space\\ \end{array}
We can see a repeating pattern emerging, that no matter how many 0‘s we add on to the divisor, we will keep getting 6‘s in the answer.
So we can fully answer the sum by rounding up with the digits we already have worked out.
8.249 ÷ 3 = 2.74967
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