Rs. 260,200 is divided between Ram and Shyam and invested at 4% compound interest per annum, compounded annually. Ram's share is invested for 3 years, while Shyam's share is invested for 6 years. The amounts they each receive at the end of their respective periods are equal. What is Ram's share (the amount initially invested in his name), in rupees?

Introduction / Context:
This problem combines compound interest with algebraic reasoning. A total sum is split between two people, Ram and Shyam, and each part grows for a different number of years at the same annual compound interest rate. The condition that both final amounts are equal allows us to set up an equation and solve for Ram's initial share.

Given Data / Assumptions:

Total sum = Rs. 260,200
Ram invests his share for 3 years at 4% per annum, compounded annually
Shyam invests his share for 6 years at the same 4% per annum, compounded annually
Final amounts received by Ram and Shyam are equal
Let Ram's initial share be R, Shyam's initial share be S
Thus R + S = 260,200

Concept / Approach:
Under annual compounding at rate r, a principal P grows after t years to:
Amount = P * (1 + r)^t, where r is in decimal form. Here r = 0.04 (since 4% per annum). Ram's amount after 3 years is R * (1.04)^3, and Shyam's amount after 6 years is S * (1.04)^6. We are told these amounts are equal, so we equate them and use the fact that S = 260,200 − R to solve for R.

Step-by-Step Solution:
Step 1: Express amounts in terms of R and S. Ram's amount = R * (1.04)^3. Shyam's amount = S * (1.04)^6. Given Ram's amount = Shyam's amount: R * (1.04)^3 = S * (1.04)^6. Step 2: Simplify the equation using S = 260,200 − R. Divide both sides by (1.04)^3: R = S * (1.04)^3 = (260,200 − R) * (1.04)^3. Let k = (1.04)^3 = 1.124864. Then R = (260,200 − R) * k. Step 3: Solve for R. R = 260,200k − Rk. R + Rk = 260,200k. R(1 + k) = 260,200k. R = 260,200k / (1 + k). Substitute k = 1.124864: R ≈ 260,200 * 1.124864 / 2.124864 ≈ Rs. 137,745 (to the nearest rupee).

Verification / Alternative check:
Using R ≈ 137,745, Shyam's share S ≈ 260,200 − 137,745 = 122,455. After 3 years, Ram's amount ≈ 137,745 * (1.04)^3 ≈ Rs. 154,945. After 6 years, Shyam's amount ≈ 122,455 * (1.04)^6, which numerically matches the same figure (up to rounding). Thus the condition that both receive equal final amounts is satisfied with Ram's share approximately Rs. 137,745.

Why Other Options Are Wrong:
If Ram's share were Rs. 125,000 or Rs. 152,000 or Rs. 108,200, then after compounding for 3 years, his amount would be significantly different from Shyam's amount after 6 years, breaking the equality condition. These values are plausible-looking but do not satisfy the precise compound interest relationship derived above.

Common Pitfalls:
One common mistake is to equate the principals directly instead of their compounded amounts, forgetting that they grow for different durations. Another is to treat the problem as simple interest, which gives a different and incorrect division. Some students also drop precision prematurely when dealing with (1.04)^3, leading to noticeable errors in the final value of R. Careful handling of the algebra and the growth factor is essential.

Final Answer:
Ram's initial share is approximately Rs. 137,745.