If the simple interest on a sum of money for 15 months at $${\text{7}}\frac{1}{2}$$ % per annum exceeds the simple interest on the same sum for 8 months at $${\text{12}}\frac{1}{2}$$ % per annum by Rs. 32.50, then the sum of money ( In Rs.) is ?

$$\eqalign{ & {{\text{T}}_1} = 15\operatorname{months} \cr & \,\,\,\,\,\, = \frac{{15}}{{12}}years \cr & {R_1} = 7\frac{1}{2}\% = \frac{{15}}{2}\% \cr & {{\text{T}}_2} = 8\operatorname{months} \cr & \,\,\,\,\,\,\, = \frac{8}{{12}}years \cr & {{\text{R}}_2} = 12\frac{1}{2}\% = \frac{{25}}{2}\% \cr & {\text{Let the principal}} = {\text{P}} \cr & {\text{According to the question,}} \cr & \Leftrightarrow \frac{{{\text{P}} \times {\text{15}} \times {\text{15}}}}{{12 \times 2 \times 100}} - \frac{{{\text{P}} \times 25 \times 8}}{{12 \times 2 \times 100}} = 32.50 \cr & \Rightarrow \frac{{225{\text{P}}}}{{2400}} - \frac{{200{\text{P}}}}{{2400}} = 32.50 \cr & \Rightarrow \frac{{25{\text{P}}}}{{2400}} = 32.50 \cr & \Rightarrow {\text{P = Rs 3120}} \cr & {\text{Hence required principal}} \cr & {\text{ = Rs}}{\text{. 3120}} \cr} $$