Division at an introductory level usually focuses mostly on examples that feature dividing larger numbers by smaller numbers.
Sometimes in Math though we can have situations where we look to divide a number by a larger number also.
We could have for example.
2 \div 4, which can also be presented as \frac{2}{4}.
It’s true that a calculator could quickly tell us the answer. 2 \div 4 = 0.5
But we want to know how to solve such sums by hand.
We can do so by using long division.
Divide a Number by a Larger Number
with Long Division
For 2 \div 4.
1)
We begin by setting up the division as usual.
\begin{array}{r} \\[-2pt] 4 {\bf{|}} {\overline{\space 2 \space\space\space\space\space\space\space\space}} \end{array}
2)
The next step is to change the form of the dividend number, her 2.
We can rewrite 2 as a decimal number, and this doesn’t alter the value of the number.
Three zeroes is a good amount to use in our new decimal form, though we may end up not needing to use all of them in the division sum.
2 = 2.000
Note that we have to also put a decimal point in the same place above the dividend.
In order to account for there being a decimal point in the final answer that we achieve.
\begin{array}{r} . \space\space\space\space\space\space\space\space \\[-2pt] 4 {\bf{|}} {\overline{\space 2.000 \space\space}} \end{array}
3)
We then look to carry out the required long division to give us a solution.
\begin{array}{r} 0. \space\space\space\space\space\space \\[-2pt] 4 {\bf{|}} {\overline{2.000}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space \\ 2 \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} 0.5 \space\space\space\space \\[-2pt] 4 {\bf{|}} {\overline{2.000}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space \\ 2\space0 \space\space\space\space \\ {\text{--}}\space \underline{2\space0} \space\space\space\space \\ 0 \space\space\space\space \end{array}
As a result of zero is obtained after the second phase of multiplication and subtraction.
The required long division is thus complete, and we have the correct answer to the sum.
2 \div 5 = 0.4
Examples
1.1
Solve 4 \div 16.
Solution
\begin{array}{r} . \space\space\space\space\space\space\space\space \\[-2pt] 16 {\bf{|}} {\overline{\space 4.000 \space\space}} \end{array}
\begin{array}{r} 0. \space\space\space\space\space\space\space \\[-2pt] 16 {\bf{|}} {\overline{\space 4.000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space \\ 4\space0 \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} 0.2 \space\space\space\space\space \\[-2pt] 16 {\bf{|}} {\overline{\space 4.000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space \\ 4\space0 \space\space\space\space\space \\ {\text{--}}\space \underline{3\space2} \space\space\space\space\space \\ 8 \space\space\space\space\space\\ \space\\ \space \end{array} => \begin{array}{r} 0.25 \space\space\space \\[-2pt] 16 {\bf{|}} {\overline{\space 4.000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space \\ 4\space0 \space\space\space\space\space \\ {\text{--}}\space \underline{3\space2} \space\space\space\space\space \\ 80 \space\space\space\\ {\text{--}}\space \underline{80} \space\space\space \\ 0 \space\space\space \end{array}
\underline{4 \div 16 = 0.25}
1.2
Solve 16 \div 50.
Solution
\begin{array}{r} . \space\space\space\space\space\space\space\space \\[-2pt] 50 {\bf{|}} {\overline{\space 16.000 \space\space}} \end{array}
\begin{array}{r} 0. \space\space\space\space\space\space\space \\[-2pt] 50 {\bf{|}} {\overline{\space 16.000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space \\ 16 \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 \\ \space \end{array} => \begin{array}{r} 0.3 \space\space\space\space\space \\[-2pt] 50 {\bf{|}} {\overline{\space 16.000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space \\ 16 \space\space\space\space\space\space\space\space \\ {\text{--}}\space \underline{15\space0} \space\space\space\space\space \\ 1\space0 \space\space\space\space\space\\ \space\\ \space \end{array} => \begin{array}{r} 0.32 \space\space\space \\[-2pt] 50 {\bf{|}} {\overline{\space 16.000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space \\ 16 \space\space\space\space\space\space\space\space \\ {\text{--}}\space \underline{15\space0} \space\space\space\space\space \\ 1\space00 \space\space\space\\ {\text{--}}\space \underline{1\space00} \space\space\space \\ 0 \space\space\space \end{array}
\underline{16 \div 50 = 0.32}
1.3
Solve 6 \div 16.
Solution
\begin{array}{r} . \space\space\space\space\space\space\space\space \\[-2pt] 16 {\bf{|}} {\overline{\space 6.000 \space\space}} \end{array}
\begin{array}{r} 0. \space\space\space\space\space\space\space \\[-2pt] 16 {\bf{|}} {\overline{\space 6.000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space \\ 6 \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 \\ \space \\ \space \\ \space \end{array} => \begin{array}{r} 0.3 \space\space\space\space\space \\[-2pt] 16 {\bf{|}} {\overline{\space 6.000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space \\ 6{\phantom{.}}0 \space\space\space\space\space \\ {\text{--}}\space \underline{4\space8} \space\space\space\space\space \\ 1\space2 \space\space\space\space\space\\ \space\\ \space \\ \space \\ \space \\ \space \\ \space \end{array} => \begin{array}{r} 0.37 \space\space\space \\[-2pt] 16 {\bf{|}} {\overline{\space 6.000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space \\ 6{\phantom{.}}0 \space\space\space\space\space \\ {\text{--}}\space \underline{4\space8} \space\space\space\space\space \\ 1\space20 \space\space\space\\ {\text{--}}\space \underline{1\space12} \space\space\space \\ 8 \space\space\space \\ \space \\ \space \\ \space \\ \space \end{array} => \begin{array}{r} 0.375 \space \\[-2pt] 16 {\bf{|}} {\overline{\space 6.000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space \\ 6{\phantom{.}}0 \space\space\space\space\space \\ {\text{--}}\space \underline{4\space8} \space\space\space\space\space \\ 1\space20 \space\space\space\\ {\text{--}}\space \underline{1\space12} \space\space\space \\ 80 \space \\ {\text{--}}\space \underline{80} \space \\ 0 \space \\ \space \\ \space \end{array}
\underline{6 \div 16 = 0.375}
Rounding Answers to Decimal Places
Sometimes when we divide a smaller number by a larger number, it can be required to round an answer to a certain ‘decimal place’.
Now we follow the same steps of long division seen earlier on this page.
But we do need to do one more additional division ahead of the decimal place we are rounding to.
As this extra digit tells us whether we round an answer up or round down.
This can be seen in practice in some examples.
Examples
2.1
Solve 4 \div 7 to 2 decimal places.
Solution
We’re to round to 2 decimal places, so three zeroes in the dividend is what we need.
\begin{array}{r} . \space\space\space\space\space\space\space\space \\[-2pt] 7 {\bf{|}} {\overline{\space 4.000 \space\space}} \end{array}
\begin{array}{r} 0. \space\space\space\space\space\space\space \\[-2pt] 7 {\bf{|}} {\overline{\space 4.000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space \\ 4 \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 \\ \space \\ \end{array} => \begin{array}{r} 0.5 \space\space\space\space\space \\[-2pt] 7 {\bf{|}} {\overline{\space 4.000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space \\ 4\space0 \space\space\space\space\space \\ {\text{--}}\space \underline{3\space5} \space\space\space\space\space \\ 5 \space\space\space\space\space\\ \space\\ \space \\ \space \\ \space \\ \end{array} => \begin{array}{r} 0.57 \space\space\space \\[-2pt] 7 {\bf{|}} {\overline{\space 4.000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space \\ 4\space0 \space\space\space\space\space \\ {\text{--}}\space \underline{3\space5} \space\space\space\space\space \\ 50 \space\space\space\\ {\text{--}}\space \underline{49} \space\space\space \\ 1 \space\space\space \\ \space \\ \space \\ \end{array}
Here we have reached 2 decimal places, but now another step will tell us if we are to round up or round down at the 7.
=> \begin{array}{r} 0.571 \space \\[-2pt] 7 {\bf{|}} {\overline{\space 4.000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space \\ 4\space0 \space\space\space\space\space \\ {\text{--}}\space \underline{3\space5} \space\space\space\space\space \\ 50 \space\space\space\\ {\text{--}}\space \underline{49} \space\space\space \\ 10 \space \\ {\text{--}}\space \underline{\space\space1} \space \\ 9 \space \\ \end{array}
At the third decimal place we have the number 1.
So in the answer we round down at the 2nd decimal place.
4 \div 7 = 0.57 to 2dp
2.2
Solve 4 \div 19 to 3 decimal places.
Solution
We’re to round to 2 decimal places, so three zeroes in the dividend is what we need.
\begin{array}{r} . \space\space\space\space\space\space\space\space\space\space \\[-2pt] 19 {\bf{|}} {\overline{\space 4.0000 \space\space}} \end{array}
\begin{array}{r} 0. \space\space\space\space\space\space\space\space\space \\[-2pt] 19 {\bf{|}} {\overline{\space 4.0000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 4 \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\space\space\space\\ \space \\ \space \\ \space \\ \space \\ \end{array} => \begin{array}{r} 0.2 \space\space\space\space\space\space\space \\[-2pt] 19 {\bf{|}} {\overline{\space 4.0000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 4\space0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{3\space8} \space\space\space\space\space\space\space \\ 2 \space\space\space\space\space\space\space\\ \space\\ \space \\ \space \\ \space \\ \end{array} => \begin{array}{r} 0.21 \space\space\space\space\space \\[-2pt] 19 {\bf{|}} {\overline{\space 4.0000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 4\space0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{3\space8} \space\space\space\space\space\space\space \\ 20 \space\space\space\space\space\\ {\text{--}}\space \underline{19} \space\space\space\space\space \\ 1 \space\space\space\space\space \\ \space \\ \space \\ \end{array} => \begin{array}{r} 0.210 \space\space\space \\[-2pt] 19 {\bf{|}} {\overline{\space 4.0000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 4\space0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{3\space8} \space\space\space\space\space\space\space \\ 20 \space\space\space\space\space\\ {\text{--}}\space \underline{19} \space\space\space\space\space \\ 10 \space\space\space \\ {\text{--}}\space \underline{0} \space\space\space \\ 10 \space\space\space \\ \end{array}
=> \begin{array}{r} 0.2105 \space \\[-2pt] 19 {\bf{|}} {\overline{\space 4.0000\space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 4\space0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{3\space8} \space\space\space\space\space\space\space \\ 20 \space\space\space\space\space\\ {\text{--}}\space \underline{19} \space\space\space\space\space \\ 10 \space\space\space \\ {\text{--}}\space \underline{0} \space\space\space \\ 100 \space \\ \end{array}
Ignoring remainders 19 goes into 100 five times.
So in the answer we round up at the 3rd decimal place.
4 \div 19 = 0.211 to 3dp
2.3
Solve 16 \div 58 to 4 decimal places.
Solution
We use 5 zeroes for the decimal places in the dividend in this example.
Due to a 5th step being required to tell us how to round at the 4th decimal place.
\begin{array}{r} . \space\space\space\space\space\space\space\space\space\space\space \\[-2pt] 58 {\bf{|}} {\overline{\space 16.00000 \space}} \end{array}
\begin{array}{r} 0. \space\space\space\space\space\space\space\space\space\space\space \\[-2pt] 58 {\bf{|}} {\overline{\space 16.00000 \space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space\space\space \\ 16 \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\space\space\space\space\space\\ \space \\ \space \\ \space \\ \space \\ \end{array} => \begin{array}{r} 0.2 \space\space\space\space\space\space\space\space\space \\[-2pt] 58 {\bf{|}} {\overline{\space 16.00000 \space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space\space\space \\ 16\space0 \space\space\space\space\space\space\space\space\space \\ {\text{--}}\space \underline{11\space6} \space\space\space\space\space\space\space\space\space \\ 4\space4 \space\space\space\space\space\space\space\space\space\\ \space\\ \space \\ \space \\ \space \\ \end{array} => \begin{array}{r} 0.27 \space\space\space\space\space\space\space \\[-2pt] 58 {\bf{|}} {\overline{\space 16.00000 \space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space\space\space \\ 16\space0 \space\space\space\space\space\space\space\space\space \\ {\text{--}}\space \underline{11\space6} \space\space\space\space\space\space\space\space\space \\ 4\space40 \space\space\space\space\space\space\space\\ {\text{--}}\space \underline{4\space06} \space\space\space\space\space\space\space \\ 34 \space\space\space\space\space\space\space \\ \space \\ \space \\ \end{array} => \begin{array}{r} 0.275 \space\space\space\space\space \\[-2pt] 58 {\bf{|}} {\overline{\space 16.00000 \space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space\space\space \\ 16\space0 \space\space\space\space\space\space\space\space\space \\ {\text{--}}\space \underline{11\space6} \space\space\space\space\space\space\space\space\space \\ 4\space40 \space\space\space\space\space\space\space\\ {\text{--}}\space \underline{4\space06} \space\space\space\space\space\space\space \\ 340 \space\space\space\space\space \\ {\text{--}}\space \underline{290} \space\space\space\space\space \\ 50 \space\space\space\space\space \\ \end{array}
=> \begin{array}{r} 0.2758 \space\space\space \\[-2pt] 58 {\bf{|}} {\overline{\space 16.00000 \space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space\space\space \\ 16\space0 \space\space\space\space\space\space\space\space\space \\ {\text{--}}\space \underline{11\space6} \space\space\space\space\space\space\space\space\space \\ 4\space40 \space\space\space\space\space\space\space\\ {\text{--}}\space \underline{4\space06} \space\space\space\space\space\space\space \\ 340 \space\space\space\space\space \\ {\text{--}}\space \underline{290} \space\space\space\space\space \\ 500 \space\space\space \\ {\text{--}}\space \underline{464} \space\space\space \\ 36 \space\space\space \end{array} => \begin{array}{r} 0.27586 \space \\[-2pt] 58 {\bf{|}} {\overline{\space 16.00000 \space}} \\ {\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space\space\space \\ 16\space0 \space\space\space\space\space\space\space\space\space \\ {\text{--}}\space \underline{11\space6} \space\space\space\space\space\space\space\space\space \\ 4\space40 \space\space\space\space\space\space\space\\ {\text{--}}\space \underline{4\space06} \space\space\space\space\space\space\space \\ 340 \space\space\space\space\space \\ {\text{--}}\space \underline{290} \space\space\space\space\space \\ 500 \space\space\space \\ {\text{--}}\space \underline{464} \space\space\space \\ 360 \space \end{array}
Ignoring any remainders 58 will go into 360 six times.
So in the answer we round up at the 4th decimal place.
16 \div 58 = 0.2759 to 4dp
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