Division of Decimal Numbers

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

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:	$3\;\mathrm{ l}:0{,}2 \;\mathrm{ l}$
You may convert the units in order to compute with integers	$3000\;\mathrm{ml}:200\;\mathrm{ml}$
Now, you may solve the exercise as usual. Answer: You can fill 15 glasses.	$\def\arraystretch{1.25} \begin{array}{l}\hphantom{-}3000\;\mathrm{ml}:200\;\mathrm{ml}=15\\\underline{-200}\\\hphantom{-}1000\\\underline{-1000}\\\hphantom{-100}0\end{array}$

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 $1$ , the result is (in absolute value) greater than the dividend. For instance, in the above example, $15$ is greater than $3$ .

Example

Description	Calculation
If necessary, add zeros to the decimal places, then move the decimal points and you can calculate as usual.	$\def\arraystretch{1.25} \begin{array}{l}\hphantom{=}137:0{,}25\\=137.00:0.25\\=13700:25\end{array}$

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.	$\phantom{=}34.765:4.09\\=3476.5:409\\=34765:4090$
First way	$\def\arraystretch{1.25} \begin{array}{l}\hphantom{-}34765.0:4090=8.5\\\underline{-32720}\\\hphantom{-3}20450\\\hphantom{3}\underline{-20450}\\\hphantom{-20450}0\end{array}$
Second way	$\def\arraystretch{1.25} \begin{array}{l}\hphantom{-}3476.5:409=8.5\\\underline{-3272}\\\hphantom{-3}2045\\\hphantom{3}\underline{-2045}\\\hphantom{-2045}0\end{array}$

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 $10,\ 100,\ 1000,\ \ldots$ . You only need to shift the decimal mark to the left, once for each zero the denominator contains.

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

Division by 0.1, 0.01, etc.

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

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

Decimal number as dividend

Example

Decimal number as divisor

Example

Division of two decimal numbers

Example

Special cases

Division by 10, 100, etc.

Division by 0.1, 0.01, etc.

Übungsaufgaben: Division of Decimal Numbers