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1. 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 profit. For how long was Y's money used in the business ?

A. 2 months

B. 3 months

C. 4 months

D. 5 months

Solution:

\begin{aligned} Let  the total profit be Rs . z \\ Then, \\ Y's share = Rs . \frac{2 z}{5} \\ X's share = Rs . (z - \frac{2 z}{5}) \\ = Rs . \frac{3 z}{5} \\ ∴ X : Y \\ = \frac{3 z}{5} : \frac{2 z}{5} \\ = 3 : 2 \end{aligned}

Let the total capital be Rs. x and
Suppose Y's money was used for y months
Then,

\begin{aligned} \Rightarrow \frac{\frac{1}{3} x \times 9}{\frac{2}{3} x \times y} = \frac{3}{2} \\ \Rightarrow 18 x = 6 x y \\ \Rightarrow y = 3 \end{aligned}

Hence, Y's money was used for 3 months.

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