A certain sum is to be divided between A and B so that after 5 years the amount received by A is equal to the amount received by B after 7 years. The rate of interest is 10% per annum, compounded annually. Find the ratio of the amounts invested by A and B.

Introduction:
This compound interest question asks you to divide an investment between two people such that their future amounts, received at different times, are equal. The key is to use the equality of future values and the compound interest formula to derive a ratio of their present investments, without needing the actual total sum.

Given Data / Assumptions:

• A invests an amount that will accumulate for 5 years.
• B invests an amount that will accumulate for 7 years.
• Rate of interest r = 10% per annum, compounded annually.
• The amounts received by A after 5 years and by B after 7 years are equal.
• We must find the ratio of the present investments of A and B.

Concept / Approach:
Let A invest an amount a and B invest an amount b. Under compound interest, A's amount after 5 years is a * (1.10)^5 and B's amount after 7 years is b * (1.10)^7. Equating these and simplifying yields a / b = (1.10)^2. Because 1.10 = 11/10, (1.10)^2 = 121/100, so the ratio a : b is 121 : 100.

Step-by-Step Solution:
Let A's present investment be a and B's present investment be b.A's amount after 5 years: A5 = a * (1 + 10/100)^5 = a * (1.1)^5.B's amount after 7 years: B7 = b * (1.1)^7.Given A5 = B7, so a * (1.1)^5 = b * (1.1)^7.Divide both sides by (1.1)^5: a = b * (1.1)^2.So a / b = (1.1)^2 = (11/10)^2 = 121/100.Therefore, the ratio of their investments is a : b = 121 : 100.

Verification / Alternative Check:
You may use assumed values such as a = 121x and b = 100x. Then A's amount after 5 years is 121x * (1.1)^5 and B's amount after 7 years is 100x * (1.1)^7. B's amount can be rewritten as 100x * (1.1)^5 * (1.1)^2 = 121x * (1.1)^5, so both amounts are indeed equal, confirming that 121 : 100 is correct.

Why Other Options Are Wrong:
125 : 100 and 100 : 125: These ratios would imply a different power relationship, not arising from (1.1)^2.100 : 121: This is simply the inverse of the correct ratio and would make B's present investment higher than A's, which contradicts the condition for equal future values at different times.11 : 10: This corresponds to a single factor of 1.1, whereas we require (1.1)^2 for the 2 year difference in time.

Common Pitfalls:
Some students mistakenly take the ratio of times 5 : 7 as the ratio of investments or use linear instead of exponential factors for compound interest. Correct handling of powers is crucial: the person whose money is invested for fewer years must start with a larger principal to catch up in future value.

Final Answer:
The required ratio of amounts invested by A and B is 121 : 100.