If the annual rate of simple interest increases from 10% to $12 \frac{1}{2}$%, a man's yearly income increases by ₹ 1250. His principal (in ₹) is (S.S.C., 2004)

### Concept & Direct Effect of Rate Change A change in the annual interest rate directly translates to a proportional change in the annual interest income. The extra income generated is exactly the simple interest calculated using the difference in the rates. Formula for Difference in Simple Interest: $$\Delta SI = \frac{P \times \Delta R \times T}{100}$$ ### Step-by-Step Solution - **Given:** - Initial Rate $R_1 = 10\%$ - New Rate $R_2 = 12 \frac{1}{2}\% = 12.5\%$ - Increase in rate $\Delta R = 12.5\% - 10\% = 2.5\%$ - Increase in yearly income $\Delta SI = 1250$ - Time $T = 1$ year (since it is "yearly income") - **Calculation:** - $1250 = \frac{P \times 2.5 \times 1}{100}$ - $1250 = \frac{2.5P}{100}$ - $P = \frac{1250 \times 100}{2.5}$ - $P = \frac{125000}{2.5}$ - $P = 50000$ ### Exam Strategy & Shortcut The increase in rate is $2.5\%$. This means $2.5\%$ of the principal equals $1250$. To find $100\%$ of the principal quickly, multiply both sides by $40$ (since $2.5 \times 4 = 10$, so $2.5 \times 40 = 100$). $1250 \times 40 = 50000$. ### Common Pitfall Misreading the fractional rate $12 \frac{1}{2}\%$ as $12.12\%$ or similar incorrect decimal conversions. Always convert mixed fraction percentages to clean decimals ($12.5\%$) or improper fractions ($25/2\%$). ### Final Answer Therefore, the correct answer is **₹ 50,000**.