Skip to content or footer

Division of Decimal Numbers

The difficulty in dividing two decimal numbers lies in the positioning of the comma in the result.

Decimal number as dividend

When calculating, you first ignore the decimal point and add it to the result as soon as you "exceed" it.

You can also make a rough calculation, e.g. by rounding to full (integer) numbers. The result of the calculation is then a good indication for the position of the decimal point.

Example

Description

Calculation

As soon as you "pull down" the 4 behind the decimal point, an analogue decimal point is placed in the result.

Alternatively: 12 fits about once in 14, so the result must be about 1.

14.4:12=1.2121  24241,,0\def\arraystretch{1.25} \begin{array}{l}\hphantom{-}14.4:12=1.2\\\underline{-12}\\\hphantom{-1}\;24\\\hphantom{-}\underline{-24}\\\hphantom{-1,,}0\end{array}

Decimal number as divisor

In real life, there are often questions, which require you to divide through a decimal number.

Example: How many 0.2-liter glasses you can fill with three liters of water?

You can write this question as a division exercise:

Description

Calculation

The quotient you have to find is:

3  l:0,2  l3\;\mathrm{ l}:0{,}2 \;\mathrm{ l}

You may convert the units in order to compute with integers

3000  ml:200  ml3000\;\mathrm{ml}:200\;\mathrm{ml}

Now, you may solve the exercise as usual.

Answer: You can fill 15 glasses.

3000  ml:200  ml=15200100010001000\def\arraystretch{1.25} \begin{array}{l}\hphantom{-}3000\;\mathrm{ml}:200\;\mathrm{ml}=15\\\underline{-200}\\\hphantom{-}1000\\\underline{-1000}\\\hphantom{-100}0\end{array}

The change of the unit can be generalized and so every division with decimal number as divisor can be converted into a division of integer numbers.

The important thing is to shift the decimal point by the same number of digits in both the dividend and divisor.

When dividing by a number less than 11, the result is (in absolute value) greater than the dividend. For instance, in the above example, 15 15 is greater than 33.

Example

Description

Calculation

If necessary, add zeros to the decimal places, then move the decimal points and you can calculate as usual.

=137:0,25=137.00:0.25=13700:25\def\arraystretch{1.25} \begin{array}{l}\hphantom{=}137:0{,}25\\=137.00:0.25\\=13700:25\end{array}

Division of two decimal numbers

If both the dividend and divisor are decimal numbers (with at least one decimal place not equal to zero), the division can be reduced to either the first case (decimal number as a dividend) or to the division of two integers. Just "move the decimal points" as above.

Example

Description

Calculation

You can calculate the result with both

transformations by dividing in writing.

=34.765:4.09=3476.5:409=34765:4090\phantom{=}34.765:4.09\\=3476.5:409\\=34765:4090

First way

34765.0:4090=8.532720320450320450204500\def\arraystretch{1.25} \begin{array}{l}\hphantom{-}34765.0:4090=8.5\\\underline{-32720}\\\hphantom{-3}20450\\\hphantom{3}\underline{-20450}\\\hphantom{-20450}0\end{array}

Second way

3476.5:409=8.53272320453204520450\def\arraystretch{1.25} \begin{array}{l}\hphantom{-}3476.5:409=8.5\\\underline{-3272}\\\hphantom{-3}2045\\\hphantom{3}\underline{-2045}\\\hphantom{-2045}0\end{array}

In the second calculation, you can see that you may have to add one (or more) decimal places with zeros.

Special cases

Division by 10, 100, etc.

A simple special case is the division of a decimal number by 10, 100, 1000, 10,\ 100,\ 1000,\ \ldots. You only need to shift the decimal mark to the left, once for each zero the denominator contains.

Example: 3.4:100=0.34:10=0.0343.4:100=0.34:10=0.034

Division by 0.1, 0.01, etc.

Quite similar is the division of a decimal number by 0.1,0.01,0.001,0.1{,}0.01{,}0.001,\dots . Here, you shift the decimal mark again, only this time to the right by one decimal place for each zero in the denominator.

Example: 3.4:0.01=34:0.1=3403.4:0.01=34:0.1=340

Exercises


This content is licensed under
CC BY-SA 4.0Info