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