X and Y are partners in a business. X contributed 1/3 of the capital for 9 months and Y received 2/5 of the profit. For how long was Y's money used in the business ?

Answer: Option B

$$\begin{array}{rl}& \text{Let the total profit be Rs}\text{.}z\\ & \text{Then,}\\ & \text{Y’s share}=\text{Rs}\text{.}\frac{2z}{5}\\ & \text{X’s share}=\text{Rs}\text{.}\left(z-\frac{2z}{5}\right)\\ & =\text{Rs}.\frac{3z}{5}\\ & \therefore \text{X}:\text{Y}\\ & =\frac{3z}{5}:\frac{2z}{5}\\ & =3:2\end{array}$$
Let the total capital be Rs. x and
Suppose Y's money was used for y months
Then,
$$\begin{array}{rl}& \Rightarrow \frac{\frac{1}{3}x\times 9}{\frac{2}{3}x\times y}=\frac{3}{2}\\ & \Rightarrow 18x=6xy\\ & \Rightarrow y=3\end{array}$$
Hence, Y's money was used for 3 months.