A garrison had provision for a certain number of days. After 10 days, $$\frac{1}{5}$$ of the men desert and it is found that the provisions will now last just as long as before. How long was that ?

Initially, Let there be x men having food for y days
After 10 days, x men had food for days (y - 10)
Also, $$\left( {x - \frac{x}{5}} \right)$$   men had food for y days
$$\eqalign{ & \therefore \,x\left( {y - 10} \right) = \frac{{4x}}{5} \times y \cr & \Leftrightarrow 5xy - 50x = 4xy \cr & \Leftrightarrow xy - 50x = 0 \cr & \Leftrightarrow x\left( {y - 50} \right) = 0 \cr & \Leftrightarrow y - 50 = 0 \cr & \Leftrightarrow y = 50 \cr} $$