In a regiment the ratio between the number of officers to soldiers was 3 : 31 before. In a battle 6 officers and 22 soldiers were killed and the ratio become 1 : 13, the number of officers in the regiment before battle was = ?

Let the number of officers and soldiers = 3x, 31x
According to the question,
$$\eqalign{ & \Rightarrow \frac{{3x - 6}}{{31x - 22}} = \frac{1}{{13}} \cr & \Rightarrow 39x - 78 = 31x - 22 \cr & \Rightarrow 8x = 56 \cr & \Rightarrow x = 7 \cr} $$
So, number of officer before
= 3x = 3 × 7 = 21