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?

Solution(By Examveda Team)

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