X and Y are partners in a business. X contributed $\frac{1}{3}$ of the capital for 9 months and Y received $\frac{2}{5}$ of the profits. For how long was Y's money used in the business? (M.A.T., 2010)
(a)2 months
(b)3 months
(c)4 months
(d)5 months
Answer
Answer (as printed): B
Explanation
Let the total profit be ₹$z$. Then, Y's share $=₹\left(\frac{2z}{5}\right)$, X's share $=₹\left(z-\frac{2z}{5}\right)=₹\left(\frac{3z}{5}\right)$. $\therefore X:Y=\frac{3z}{5}:\frac{2z}{5}=3:2$. Let the total capital be ₹$x$ and suppose Y's money was used for $y$ months. Then, $\frac{\frac{1}{3}x\times9}{\frac{2}{3}x\times y}=\frac{3}{2} \Rightarrow 18x=6xy \Rightarrow y=3$. Hence, Y's money was used for 3 months.
Explanation as extracted from the printed page; notation may be imperfect.