A body of 7300 troops is formed of 4 battalions so that $$\frac{1}{2}$$ of the first, $$\frac{2}{3}$$ of the second, $$\frac{3}{4}$$ of the third and $$\frac{4}{5}$$ of the fourth are all composed of the same number of men. How many men are there in the second battalion?

Explanation:-

Let the number of men in the 1st, 2nd, 3rd and 4th battalions be x, y, z and t respectively.
Then,
$$\eqalign{ & \frac{1}{2}x = \frac{2}{3}y = \frac{3}{4}z = \frac{4}{5}t \cr & \Rightarrow x = \frac{4}{3}y, \cr & \,\,\,\,\,\,\,\,z = \frac{8}{9}y, \cr & \,\,\,\,\,\,\,\,\,t = \frac{5}{6}y{\text{ }} \cr & {\text{Now,}} \cr & x + y + z + t \cr & = 7300 \cr & \Rightarrow \frac{4}{3}y + y + \frac{8}{9}y + \frac{5}{6}y \cr & \,\,\,\,\,\,\, = 7300 \cr & \Rightarrow \frac{{24y + 18y + 16y + 15y}}{{18}} \cr & \,\,\,\,\,\,\,\, = 7300 \cr & \Rightarrow 73y = 7300 \times 18 \cr & \Rightarrow y = 1800 \cr}$$