A person lent out a certain sum on simple interest and the same sum on compound interest at a certain rate of interest per annum. He noticed that the ratio between the difference of compound interest and simple interest of 3 years and that of 2 years is 25 : 8. The rate of interest per annum is

### Concept & Ratio of 3-Year to 2-Year CI-SI Differences For a principal $P$ at rate $R\%$, the difference between CI and SI for 2 years ($D_2$) and 3 years ($D_3$) follows a direct algebraic ratio. $$D_2 = P\left(\frac{R}{100}\right)^2$$ $$D_3 = P\left(\frac{R}{100}\right)^2 \times \left(3 + \frac{R}{100}\right)$$ $$\frac{D_3}{D_2} = 3 + \frac{R}{100}$$ ### Step-by-Step Solution 1. **Equate the formula to the given ratio:** The ratio is given as $25 : 8$. $\frac{D_3}{D_2} = \frac{25}{8}$ 2. **Substitute into the standard formula:** $3 + \frac{R}{100} = \frac{25}{8}$ 3. **Solve for R:** $\frac{R}{100} = \frac{25}{8} - 3$ $\frac{R}{100} = \frac{25 - 24}{8}$ $\frac{R}{100} = \frac{1}{8}$ $R = \frac{100}{8} = 12.5$ 4. **Format the answer:** $12.5\% = 12\frac{1}{2}\%$ ### Exam Strategy & Shortcut Memorize the relationship $\frac{D_3}{D_2} = 3 + \frac{R}{100}$ for CI-SI difference problems. When you see a ratio like $\frac{25}{8}$, immediately split it as $3 + \frac{1}{8}$. This immediately tells you that $\frac{R}{100} = \frac{1}{8} = 12.5\%$. ### Common Pitfall Trying to solve this by expanding the full $(1+R/100)^3$ binomial formula manually from scratch will consume minutes. Recognize the $D_3/D_2$ shortcut pattern. ### Final Answer Therefore, the correct answer is **$12\frac{1}{2}\%$**.