Rs. 5887 is divided between Shyam and Ram such that Shyam receives a share which, at the end of 9 years, is equal to Ram's share at the end of 11 years, when both shares earn 5 percent compound interest per annum. What is Shyam's share?

Introduction / Context:
This question is about dividing a sum between two people, Shyam and Ram, so that their future amounts under compound interest become equal at different times. Shyam holds his share for 9 years and Ram for 11 years, both at 5 percent compound interest per annum. Their amounts after these periods are equal. We must use this condition to determine Shyam's initial share, given that together they receive Rs. 5887.

Given Data / Assumptions:

• Total sum to be divided = Rs. 5887.
• Let Shyam's share be S and Ram's share be R.
• S + R = 5887.
• Each share earns 5 percent per annum compound interest.
• Shyam's amount after 9 years equals Ram's amount after 11 years.

Concept / Approach:
The compound amount after n years at rate r percent is: A = P * (1 + r/100)^n For Shyam, the amount after 9 years is S * (1.05)^9. For Ram, the amount after 11 years is R * (1.05)^11. Given these amounts are equal, we have: S * (1.05)^9 = R * (1.05)^11 From this we get a relation between S and R, and combined with S + R = 5887 we can solve for S and R.

Step-by-Step Solution:
Given S * (1.05)^9 = R * (1.05)^11 Divide both sides by (1.05)^9: S = R * (1.05)^2 (1.05)^2 = 1.1025 So S = 1.1025 * R Also, S + R = 5887 Substitute S = 1.1025 * R into S + R = 5887: 1.1025 * R + R = 5887 (1.1025 + 1) * R = 5887 2.1025 * R = 5887 R = 5887 / 2.1025 ≈ 2800 Then S = 5887 - 2800 = 3087 Therefore, Shyam's share is Rs. 3087

Verification / Alternative check:
We can verify by forwarding each share. If S = 3087, then R = 5887 - 3087 = 2800. Shyam's amount after 9 years is: 3087 * (1.05)^9 Ram's amount after 11 years is: 2800 * (1.05)^11 Since S = R * (1.05)^2, we know: S * (1.05)^9 = R * (1.05)^2 * (1.05)^9 = R * (1.05)^11 So the two final amounts are indeed equal, which confirms that the division S = 3087 and R = 2800 satisfies the condition exactly.

Why Other Options Are Wrong:
Values such as 3567, 3452, 3544, or 3000 for Shyam's share do not satisfy both the total sum and the equality condition when compounded for 9 and 11 years respectively. If we use any of these as S and compute R = 5887 - S, the final amounts after 9 and 11 years at 5 percent will not match. Only S = 3087 and R = 2800 produce equal future amounts under the given interest rate and time periods.

Common Pitfalls:
Confusing compound interest with simple interest and trying to use linear relations is a common error. Some students may also attempt to expand (1.05)^9 and (1.05)^11 numerically rather than using the simpler relation that comes from dividing the equations. Misinterpreting which share belongs to which person can also cause mistakes. Using the ratio S / R = (1.05)^2 and the total sum equation side by side provides a clean and reliable method.

Final Answer:
Shyam's share out of Rs. 5887 is Rs. 3087.