Divide Rs. 1301 between A and B so that the amount of A after 7 years is equal to the amount of B after 9 years, the rate of interest being 4% per annum compounded annually. What is B's share?

Introduction / Context:
This question involves distributing a given sum between two people such that their respective amounts grow under compound interest and become equal at different future times. The rate is the same for both, but the compounding periods differ. We must use the compound interest formula on both parts and then solve for the share belonging to one of them.

Given Data / Assumptions:

• Total sum to be divided = Rs. 1301.
• A receives Rs. x, B receives Rs. 1301 - x initially.
• Rate of interest r = 4% per annum compounded annually.
• Time for A = 7 years, time for B = 9 years.
• The amounts of A and B after their respective periods are equal.
• We must find B's share.

Concept / Approach:

Let x be the share of A. A's amount after 7 years is x * (1.04)^7. B's share is 1301 - x, and B's amount after 9 years is (1301 - x) * (1.04)^9. The condition that these amounts are equal gives an equation in x. We solve this equation to find x and then compute B's share as 1301 - x.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

Values such as Rs. 626 and Rs. 627 would not yield an exact equality of the future amounts when subjected to 7 and 9 years of compounding at 4%. Rs. 286 is far from the correct value and clearly cannot satisfy the equality condition. Only Rs. 625 produces the required equality.

Common Pitfalls:

A frequent mistake is to equate the initial shares instead of the future amounts, or to forget to apply different exponents for 7 and 9 years. Another pitfall is approximating the growth factor too roughly, which can obscure the neat result that A's share becomes a simple integer.

Final Answer:

B's share is Rs. 625.