A father left a will of ₹ 16400 for his two sons aged 17 and 18 years. They must get equal amounts when they are 20 years, at 5% compound interest. Find the present share of the younger son.

### Concept & Equating Future Amounts When a total sum is divided such that future amounts are equal, the present shares must be inversely proportional to their respective compound interest growth factors. $$Amount = P\left(1 + \frac{R}{100}\right)^T$$ ### Step-by-Step Solution 1. **Identify the time periods:** Younger son (age 17) gets interest for $20 - 17 = 3$ years. Let his share be $A$. Elder son (age 18) gets interest for $20 - 18 = 2$ years. Let his share be $B$. 2. **Set up the equality:** Both must have equal amounts at age 20. The rate $R = 5\%$. $A \times (1 + 0.05)^3 = B \times (1 + 0.05)^2$ $A \times (1.05)^3 = B \times (1.05)^2$ 3. **Find the ratio of their shares:** Dividing both sides by $(1.05)^2$: $A \times 1.05 = B$ $B = A \times \frac{105}{100} = A \times \frac{21}{20}$ So, the ratio $A : B = 20 : 21$. 4. **Calculate the younger son's share ($A$):** Total sum = $A + B = 16400$ Total ratio units = $20 + 21 = 41$ 1 unit = $\frac{16400}{41} = 400$ Younger son's share ($A$) = $20 \times 400 = 8000$. ### Exam Strategy & Shortcut Difference in time = 1 year. This means the elder son's share is just the younger son's share multiplied by $(1 + R/100)$ once. Ratio of younger to elder = $100 : (100 + R) = 100 : 105 = 20 : 21$. Younger share = $\frac{20}{(20+21)} \times 16400 = \frac{20}{41} \times 16400 = 8000$. ### Common Pitfall A common error is confusing which son gets the larger principal. The younger son has *more* time to earn interest, so his initial principal ($A$) must be *smaller* to reach the same final amount. ### Final Answer Therefore, the correct answer is **₹ 8000**.