A sum of money invested at compound interest amounts to ₹ 4624 in 2 years and to ₹ 4913 in 3 years. The sum of money is :

### Concept & Deriving Principal from Consecutive Amounts First, find the rate of interest using the difference between consecutive years' amounts. Then, trace backward using the compound interest formula or multipliers to find the original principal. $$Amount = P\left(1 + \frac{R}{100}\right)^T$$ ### Step-by-Step Solution 1. **Find the interest for the 3rd year:** Amount after 2 years ($A_2$) = ₹ 4624 Amount after 3 years ($A_3$) = ₹ 4913 Interest = $4913 - 4624 = 289$ 2. **Calculate the rate of interest ($R$):** $R = \left(\frac{289}{4624}\right) \times 100$ Notice that $289 \times 16 = 4624$. $R = \left(\frac{1}{16}\right) \times 100 = 6.25\%$ 3. **Find the principal ($P$):** The multiplier for each year is $\left(1 + \frac{1}{16}\right) = \frac{17}{16}$. $P \times \left(\frac{17}{16}\right)^2 = A_2$ $P \times \left(\frac{17}{16}\right)^2 = 4624$ $P \times \left(\frac{289}{256}\right) = 4624$ $P = \frac{4624 \times 256}{289}$ $P = 16 \times 256 = 4096$ ### Exam Strategy & Shortcut Recognize squares and cubes. $4913$ is $17^3$. $4624$ is $68^2$, but more usefully, the multiplier is $\frac{4913}{4624} = \frac{17}{16}$. Since $A_2 = P \times \left(\frac{17}{16}\right)^2$, then $P = A_2 \times \left(\frac{16}{17}\right)^2 = 4624 \times \frac{256}{289} = 4096$. ### Common Pitfall A common pitfall is stopping after finding the rate of interest, or making calculation errors when squaring fractions like $\frac{17}{16}$. Always carry out the final reverse calculation step carefully. ### Final Answer Therefore, the correct answer is **₹ 4096**.