Dividing Smaller Numbers by Larger Numbers (2024)

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 go about dividing smaller numbers by larger numbers also.

We could have for example.

2 \div 5, which can also be presented as \frac{2}{5}.

It’s true that a calculator could quickly tell us the answer.

2 \div 5 = 0.4

But we want to know how to solve such sums by hand.
We can do so by using long division.

Dividing Smaller Numbers by
Larger Numbers with Long Division

For 2 \div 5.

1)
We first set the division up as normal.
\begin{array}{r}\\[-2pt]5 {\bf{|}} {\overline{\space 2 \space\space\space\space\space\space\space\space}} \end{array}

2)
Next we want to change the form of the smaller dividend, 2.
We rewrite 2 as a decimal number, which doesn’t change the value of the number at all.

Four zeroes is a fine standard amount to use, even if we may or may not need to use all of them in the division.

But we also have to place a decimal point in the same place above.
To account for there also being a decimal point in the answer we will obtain.

\begin{array}{r}. \space\space\space\space\space\space\space\space\space\space \\[-2pt]5 {\bf{|}} {\overline{\space 2.0000 \space\space}} \end{array}

3)
We then carry out the required long division to obtain a solution.

\begin{array}{r}0. \space\space\space\space\space\space\space\space \\[-2pt]5 {\bf{|}} {\overline{2.0000}} \\{\text{--}}\space \underline{ 0 \space\space\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\space\space\space\space\space\space \end{array} => \begin{array}{r}0.4 \space\space\space\space\space\space \\[-2pt]5 {\bf{|}} {\overline{2.0000}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space \\ 2\space0 \space\space\space\space\space\space \\ {\text{--}}\space \underline{2\space0} \space\space\space\space\space\space \\ 0 \space\space\space\space\space\space\end{array}

With a result of zero obtained after the second round of multiplication and subtraction.
The long division is complete, and we have the correct answer.

2 \div 5 = 0.4

Examples

(1.1)

Solve

12 \div 25.

Solution

\begin{array}{r}. \space\space\space\space\space\space\space\space\space\space \\[-2pt]25 {\bf{|}} {\overline{\space 12.0000 \space\space}} \end{array}

\begin{array}{r}0. \space\space\space\space\space\space\space\space\space \\[-2pt]25 {\bf{|}} {\overline{\space 12.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 12 \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}0.4 \space\space\space\space\space\space\space \\[-2pt]25 {\bf{|}} {\overline{\space 12.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 12\space0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{10\space0} \space\space\space\space\space\space\space \\ 2\space0 \space\space\space\space\space\space\space\\ \space\\ \space\end{array} => \begin{array}{r}0.48 \space\space\space\space\space \\[-2pt]25 {\bf{|}} {\overline{\space 12.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 12\space0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{10\space0} \space\space\space\space\space\space\space \\ 2\space00 \space\space\space\space\space \\ {\text{--}}\space \underline{2\space00} \space\space\space\space\space \\ 0 \space\space\space\space\space\end{array}

\underline{12 \div 25 = 0.48}

(1.2)

Solve

14 \div 56.

Solution

\begin{array}{r}. \space\space\space\space\space\space\space\space\space\space \\[-2pt]56 {\bf{|}} {\overline{\space 14.0000 \space\space}} \end{array}

\begin{array}{r}0. \space\space\space\space\space\space\space\space\space \\[-2pt]56 {\bf{|}} {\overline{\space 14.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 14 \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]56 {\bf{|}} {\overline{\space 14.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 14\space0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{11\space2} \space\space\space\space\space\space\space \\ 2\space8 \space\space\space\space\space\space\space\\ \space\\ \space\end{array} => \begin{array}{r}0.25 \space\space\space\space\space \\[-2pt]56 {\bf{|}} {\overline{\space 14.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 14\space0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{11\space2} \space\space\space\space\space\space\space \\ 2\space80 \space\space\space\space\space \\ {\text{--}}\space \underline{2\space80} \space\space\space\space\space \\ 0 \space\space\space\space\space\end{array}

\underline{14 \div 56 = 0.25}

(1.3)

Solve

9 \div 16.

Solution

\begin{array}{r}. \space\space\space\space\space\space\space\space\space\space \\[-2pt]16 {\bf{|}} {\overline{\space 9.0000 \space\space}} \end{array}

\begin{array}{r}0. \space\space\space\space\space\space\space\space\space \\[-2pt]16 {\bf{|}} {\overline{\space 9.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 9 \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}0.5 \space\space\space\space\space\space\space \\[-2pt]16 {\bf{|}} {\overline{\space 9.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 9{\phantom{.}}0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{8\space0} \space\space\space\space\space\space\space \\ 1\space0 \space\space\space\space\space\space\space\\ \space\\ \space \\\space \\ \space \\\space \\ \space\end{array} => \begin{array}{r}0.56 \space\space\space\space\space \\[-2pt]16 {\bf{|}} {\overline{\space 9.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 9\space0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{8\space0} \space\space\space\space\space\space\space \\ 1\space00 \space\space\space\space\space\\ {\text{--}}\space \underline{96} \space\space\space\space\space \\ 4 \space\space\space\space\space \\\space \\ \space \\\space \\ \space\end{array} => \begin{array}{r}0.562 \space\space\space \\[-2pt]16 {\bf{|}} {\overline{\space 9.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 9\space0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{8\space0} \space\space\space\space\space\space\space \\ 1\space00 \space\space\space\space\space\\ {\text{--}}\space \underline{96} \space\space\space\space\space \\ 40 \space\space\space \\ {\text{--}}\space \underline{32} \space\space\space \\ 8 \space\space\space \\ \space \\ \space\end{array} => \begin{array}{r}0.5625 \space \\[-2pt]16 {\bf{|}} {\overline{\space 9.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 9\space0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{8\space0} \space\space\space\space\space\space\space \\ 1\space00 \space\space\space\space\space\\ {\text{--}}\space \underline{96} \space\space\space\space\space \\ 40 \space\space\space \\ {\text{--}}\space \underline{32} \space\space\space \\ 80 \space \\ {\text{--}}\space \underline{80} \space \\ 0 \space\end{array}

\underline{9 \div 16 = 0.5625}

Rounding an answer to a Decimal Place

Sometimes when dividing smaller numbers by larger numbers, it can be required to round to a specific ‘decimal place’.

We follow the same process of long division as already seen on this page.

But we would need to do one extra division beyond the decimal place we are focusing on.
As we would need to know whether we are to round up or round down.

This can be seen by looking at some examples.

Examples

(2.1)

Solve

3 \div 7 to 3 decimal places.

Solution

\begin{array}{r}. \space\space\space\space\space\space\space\space\space\space \\[-2pt]7 {\bf{|}} {\overline{\space 3.0000 \space\space}} \end{array}

\begin{array}{r}0. \space\space\space\space\space\space\space\space\space \\[-2pt]7 {\bf{|}} {\overline{\space 3.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\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\space\space\space\space\space\\ \space \\ \space \\\space \\ \space \\\end{array} => \begin{array}{r}0.4 \space\space\space\space\space\space\space \\[-2pt]7 {\bf{|}} {\overline{\space 3.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 3{\phantom{.}}0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{2\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}0.42 \space\space\space\space\space \\[-2pt]7 {\bf{|}} {\overline{\space 3.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 3{\phantom{.}}0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{2\space8} \space\space\space\space\space\space\space \\ 20 \space\space\space\space\space\\ {\text{--}}\space \underline{14} \space\space\space\space\space \\ 6 \space\space\space\space\space \\\space \\ \space \\\end{array} => \begin{array}{r}0.428 \space\space\space \\[-2pt]7 {\bf{|}} {\overline{\space 3.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 3{\phantom{.}}0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{2\space8} \space\space\space\space\space\space\space \\ 20 \space\space\space\space\space\\ {\text{--}}\space \underline{14} \space\space\space\space\space \\ 60 \space\space\space \\ {\text{--}}\space \underline{56} \space\space\space \\ 4 \space\space\space \\\end{array}

Now we are at 3 decimal places as required, but another step will let us know if we are to round up or down at the 8.

=> \begin{array}{r}0.4285 \space \\[-2pt]7 {\bf{|}} {\overline{\space 3.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 3{\phantom{.}}0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{2\space8} \space\space\space\space\space\space\space \\ 20 \space\space\space\space\space\\ {\text{--}}\space \underline{14} \space\space\space\space\space \\ 60 \space\space\space \\ {\text{--}}\space \underline{56} \space\space\space \\ 40 \space \\\end{array}

Ignoring remainders 7 goes into 40 five times.
Letting us know that in the answer we round up at the 3rd decimal place.

3 \div 7 = 0.429 to 3dp

(2.2)

Solve

8 \div 14 to 3 decimal places.

Solution

\begin{array}{r}. \space\space\space\space\space\space\space\space\space\space \\[-2pt]14 {\bf{|}} {\overline{\space 8.0000 \space\space}} \end{array}

\begin{array}{r}0. \space\space\space\space\space\space\space\space\space \\[-2pt]14 {\bf{|}} {\overline{\space 8.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 8 \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}0.5 \space\space\space\space\space\space\space \\[-2pt]14 {\bf{|}} {\overline{\space 8.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 8\space0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{7\space0} \space\space\space\space\space\space\space \\ 1\space0 \space\space\space\space\space\space\space\\ \space\\ \space \\\space \\ \space \\\end{array} => \begin{array}{r}0.57 \space\space\space\space\space \\[-2pt]14 {\bf{|}} {\overline{\space 8.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 8\space0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{7\space0} \space\space\space\space\space\space\space \\ 1\space00 \space\space\space\space\space\\ {\text{--}}\space \underline{98} \space\space\space\space\space \\ 2 \space\space\space\space\space \\\space \\ \space \\\end{array} => \begin{array}{r}0.571 \space\space\space \\[-2pt]14 {\bf{|}} {\overline{\space 8.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 8\space0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{7\space0} \space\space\space\space\space\space\space \\ 1\space00 \space\space\space\space\space\\ {\text{--}}\space \underline{98} \space\space\space\space\space \\ 20 \space\space\space \\ {\text{--}}\space \underline{14} \space\space\space \\ 6 \space\space\space \\\end{array}

=> \begin{array}{r}0.5714 \space \\[-2pt]14 {\bf{|}} {\overline{\space 8.0000\space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space \\ 8\space0 \space\space\space\space\space\space\space \\ {\text{--}}\space \underline{7\space0} \space\space\space\space\space\space\space \\ 1\space00 \space\space\space\space\space\\ {\text{--}}\space \underline{98} \space\space\space\space\space \\ 20 \space\space\space \\ {\text{--}}\space \underline{14} \space\space\space \\ 60 \space \\\end{array}

Ignoring remainders 14 goes into 60 four times.
Letting us know that in the answer we round down at the 3rd decimal place.

8 \div 14 = 0.571 to 3dp

(2.3)

Solve

14 \div 65 to 4 decimal places.

Solution

\begin{array}{r}. \space\space\space\space\space\space\space\space\space\space\space \\[-2pt]65 {\bf{|}} {\overline{\space 14.00000 \space}} \end{array}

We put 5 zeroes in the dividend this time.
As a 5th step is required to let us know which way to round at the 4th decimal place.

\begin{array}{r}0. \space\space\space\space\space\space\space\space\space\space\space \\[-2pt]65 {\bf{|}} {\overline{\space 14.00000 \space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space\space\space \\ 14 \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]65 {\bf{|}} {\overline{\space 14.00000 \space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space\space\space \\ 14\space0 \space\space\space\space\space\space\space\space\space \\ {\text{--}}\space \underline{13\space0} \space\space\space\space\space\space\space\space\space \\ 1\space0 \space\space\space\space\space\space\space\space\space\\ \space\\ \space \\\space \\ \space \\\end{array} => \begin{array}{r}0.21 \space\space\space\space\space\space\space \\[-2pt]65 {\bf{|}} {\overline{\space 14.00000 \space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space\space\space \\ 14\space0 \space\space\space\space\space\space\space\space\space \\ {\text{--}}\space \underline{13\space0} \space\space\space\space\space\space\space\space\space \\ 1\space00 \space\space\space\space\space\space\space\\ {\text{--}}\space \underline{65} \space\space\space\space\space\space\space \\ 35 \space\space\space\space\space\space\space \\\space \\ \space \\\end{array} => \begin{array}{r}0.215 \space\space\space\space\space \\[-2pt]65 {\bf{|}} {\overline{\space 14.00000 \space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space\space\space \\ 14\space0 \space\space\space\space\space\space\space\space\space \\ {\text{--}}\space \underline{13\space0} \space\space\space\space\space\space\space\space\space \\ 1\space00 \space\space\space\space\space\space\space\\ {\text{--}}\space \underline{65} \space\space\space\space\space\space\space \\ 350 \space\space\space\space\space \\ {\text{--}}\space \underline{325} \space\space\space\space\space \\ 25 \space\space\space\space\space \\\end{array}

=> \begin{array}{r}0.2152 \space\space\space \\[-2pt]65 {\bf{|}} {\overline{\space 14.00000 \space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space\space\space \\ 14\space0 \space\space\space\space\space\space\space\space\space \\ {\text{--}}\space \underline{13\space0} \space\space\space\space\space\space\space\space\space \\ 1\space00 \space\space\space\space\space\space\space\\ {\text{--}}\space \underline{65} \space\space\space\space\space\space\space \\ 350 \space\space\space\space\space \\ {\text{--}}\space \underline{325} \space\space\space\space\space \\ 250 \space\space\space \\ {\text{--}}\space \underline{125} \space\space\space \\ 55 \space\end{array} => \begin{array}{r}0.21528 \space \\[-2pt]65 {\bf{|}} {\overline{\space 14.00000 \space}} \\{\text{--}}\space \underline{ 0 \space\space\space\space}\space\space\space\space\space\space\space\space \\ 14\space0 \space\space\space\space\space\space\space\space\space \\ {\text{--}}\space \underline{13\space0} \space\space\space\space\space\space\space\space\space \\ 1\space00 \space\space\space\space\space\space\space\\ {\text{--}}\space \underline{65} \space\space\space\space\space\space\space \\ 350 \space\space\space\space\space \\ {\text{--}}\space \underline{325} \space\space\space\space\space \\ 250 \space\space\space \\ {\text{--}}\space \underline{125} \space\space\space \\ 550 \space\end{array}

Ignoring remainders 65 goes into 550 eight times.
Letting us know that in the answer we round up at the 4th decimal place.

14 \div 65 = 0.2154 to 4dp

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