The division of two decimal fractions can be rearranged into a division of two integers.
This division can be handled as usual.

Procedure

In a division, numerator and denominator can be multiplied with the same figure without changing the result. This works like expanding a fraction:

%%\frac34=\frac{3\cdot2}{4\cdot2}=\frac{3\cdot3}{4\cdot3}=\frac{3\cdot4}{4\cdot4}=\dots%%

Since multiplying with %%10%% shifts the decimal mark by one decimal place to the right, multiply both figures with %%10%% until they are integers.

Afterwards you can utilize long division.

Example:

%%15:0.2=150:2%%

%%\begin{array}{l} \hphantom{2..}\underline{\;\hphantom{.5}\color{green}{75}}\\ 2)\hphantom{-}150\\ \hphantom{2)}\underline{-150}\\ \hphantom{2),}\hphantom{100}0\\ \end{array}%%

It does not matter if numerator, denominator or both are decimal fractions. By expanding both can be turned into integers.

%%8.25:0.5=825:50%%

%%\begin{array}{l} \hphantom{50)}\underline{\;\hphantom{.5}\color{green}{16.5}}\\ 50)\hphantom{-}825\\ \hphantom{50)}\underline{-50}\\ \hphantom{50)}\hphantom{1,}325\\ \hphantom{50)}\underline{-300}\\ \hphantom{50)}\hphantom{-1}250 \end{array}%%

Special cases

  • A simple special case is the division of a decimal fraction by multiples of %%10%%. 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%%

  • Quite similar is the division of a decimal fraction by %%0.1, 0.01, 0.001,…%% . 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%%

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