Rs. 5,887 is divided between Shyam and Ram such that Shyam's share, when compounded annually at 5% for 9 years, becomes equal to Ram's share, when compounded annually at 5% for 11 years. What is Shyam's share of the original amount?

Introduction / Context:
This question involves dividing a sum of money between two people, Shyam and Ram, under conditions involving compound interest. Their shares grow at the same rate (5% per annum) but for different time periods (9 years for Shyam and 11 years for Ram), and the final amounts are equal. You must find Shyam's initial share.

Given Data / Assumptions:
- Total amount to be divided = Rs. 5,887.
- Let Shyam's share be S and Ram's share be R, with S + R = 5887.
- Annual compound interest rate = 5% per annum for both.
- Shyam's amount after 9 years = S * (1.05)^9.
- Ram's amount after 11 years = R * (1.05)^11.
- These two future amounts are equal.

Concept / Approach:
The key idea is that if two amounts grow at the same interest rate but for different times and end up equal, then their principals must be proportional to the inverse of the growth factors. Here, S * (1.05)^9 = R * (1.05)^11. We can divide both sides by (1.05)^9 to relate S and R, then use S + R = 5887 to solve for each share.

Step-by-Step Solution:
Step 1: From the equality of future values: S * (1.05)^9 = R * (1.05)^11.Step 2: Divide both sides by (1.05)^9: S = R * (1.05)^2.Step 3: Compute (1.05)^2 = 1.1025, so S = 1.1025R.Step 4: Use the total: S + R = 5887.Step 5: Substitute S = 1.1025R: 1.1025R + R = 2.1025R = 5887.Step 6: Solve for R: R = 5887 / 2.1025 = 2800.Step 7: Find S: S = 1.1025 * 2800 = 3087.Step 8: Therefore, Shyam's share is Rs. 3,087.

Verification / Alternative check:
Check the future values: Shyam's future amount after 9 years is 3087 * (1.05)^9; Ram's future amount after 11 years is 2800 * (1.05)^11. Since 2800 * (1.05)^11 = 2800 * (1.05)^2 * (1.05)^9 = (2800 * 1.1025) * (1.05)^9 = 3087 * (1.05)^9, they are indeed equal, confirming our relationship and values.

Why Other Options Are Wrong:
Values such as Rs. 3,567, Rs. 3,452 or Rs. 3,544 do not satisfy both the total sum and the equality of future values. If you plug them into S + R = 5887 and compute the compounded amounts, the final values will not match. Rs. 2,800 is Ram's share, not Shyam's.

Common Pitfalls:
Students sometimes set S * 9 equal to R * 11 (treating the comparison as simple interest), or forget that the same rate but different times implies different growth factors. Another mistake is miscomputing (1.05)^2. Always use the exponential factors correctly and remember to apply the total-sum condition S + R = 5887.

Final Answer:
Shyam's share of the original amount is Rs. 3,087.