Introduction / Context:
This question combines compound interest with ratio reasoning. Two people, Ram and Shyam, share an initial sum in such a way that the amounts they will each have after different time periods become equal. Because the rate of interest is the same for both, the principal shares are in a fixed ratio related to the growth factors over 3 and 6 years, respectively.
Given Data / Assumptions:
- Total principal P_total = Rs 260200 shared between Ram and Shyam.
- Ram invests his share for 3 years at 4% compound interest.
- Shyam invests his share for 6 years at 4% compound interest.
- Interest is compounded annually in both cases.
- The amount Ram receives after 3 years is equal to the amount Shyam receives after 6 years.
Concept / Approach:
If R is Ram's share and S is Shyam's share, then R + S = 260200. The amount Ram receives after 3 years is R * (1.04)^3, and Shyam's amount after 6 years is S * (1.04)^6. Since these amounts are equal, we have R * (1.04)^3 = S * (1.04)^6. Simplifying this equation gives the ratio R/S. Once we know the ratio, we can split the total principal accordingly and then approximate Ram's share to the nearest rupee.
Step-by-Step Solution:
Step 1: Write the equality of future amounts:
R * (1.04)^3 = S * (1.04)^6.
Step 2: Divide both sides by (1.04)^3:
R = S * (1.04)^3.
Step 3: Compute the factor:
(1.04)^3 = 1.124864 approximately.
Step 4: So R = 1.124864 * S.
Step 5: Use the total principal: R + S = 260200, so 1.124864S + S = 260200, giving 2.124864S = 260200.
Step 6: Solve for S:
S = 260200 / 2.124864 ≈ 122455.
Step 7: Then R = 1.124864 * 122455 ≈ 137745.
Verification / Alternative check:
Compute Ram's amount after 3 years: 137745 * (1.04)^3 ≈ 137745 * 1.124864 ≈ 154800 (approximate).
Compute Shyam's amount after 6 years: 122455 * (1.04)^6 ≈ 122455 * (1.124864^2) ≈ 154800 (approximate), confirming that the amounts are equal within rounding.
Why Other Options Are Wrong:
Rs 130000 or Rs 140000 do not maintain the precise equality between the two future amounts at 4% compound interest.
Rs 122455 is actually Shyam's approximate share, not Ram's.
Common Pitfalls:
Students may incorrectly treat this as a simple interest problem or may attempt to set R * 3 * 4/100 equal to S * 6 * 4/100, which is wrong because the problem specifically mentions compound interest. Another error is to assume that shares are in simple proportion of years, ignoring the exponential effect of compounding.
Final Answer:
Ram's approximate share of the original sum is Rs 137745.