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 ?

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