At what annual rate of compound interest (compounded annually) will a sum of money amount to Rs. 2,400 in 3 years and to Rs. 2,520 in 4 years?

Introduction / Context:
This problem is a classic way to find the compound interest rate when you are given two successive amounts, one after 3 years and one after 4 years. The key observation is that the ratio of the amount after 4 years to the amount after 3 years equals the growth factor for one additional year, namely (1 + r/100).

Given Data / Assumptions:
- Amount after 3 years, A3 = Rs. 2,400.
- Amount after 4 years, A4 = Rs. 2,520.
- Interest is compounded annually at a constant rate r% per annum.
- We must find r, the annual compound interest rate.

Concept / Approach:
Let P be the principal and r be the rate. Then:
- A3 = P * (1 + r/100)^3.
- A4 = P * (1 + r/100)^4.
Dividing A4 by A3 cancels P and the power 3, leaving only one factor of (1 + r/100). Thus, (A4 / A3) = (1 + r/100). We can compute this ratio directly from the amounts and deduce r.

Step-by-Step Solution:
Step 1: Compute the ratio A4 / A3 = 2520 / 2400.Step 2: Simplify: 2520 / 2400 = 252 / 240 = 21 / 20 = 1.05.Step 3: By the compound interest relationship, A4 / A3 = 1 + r/100.Step 4: So 1 + r/100 = 1.05.Step 5: Subtract 1 from both sides: r/100 = 0.05.Step 6: Multiply by 100: r = 5.Step 7: Therefore, the required annual compound interest rate is 5%.

Verification / Alternative check:
To verify, assume we know r = 5% and the amount after 3 years is Rs. 2,400. Then the amount after 4 years should be 2400 * 1.05 = 2520, which matches the given A4. This confirms that 5% is correct. We do not even need the original principal to check this relationship.

Why Other Options Are Wrong:
Any rate smaller than 5% (like 3.5% or 4%) would give an A4/A3 ratio less than 1.05. Any rate above 5% (like 6% or 6.5%) would produce a ratio higher than 1.05. Since the actual ratio is exactly 1.05, only r = 5% satisfies the compound growth pattern.

Common Pitfalls:
Some students mistakenly use simple interest logic and try to tie the difference of amounts directly to 4 years instead of 1 year. Others try to solve for principal first, which is unnecessary and can introduce extra algebra. Remember that the ratio of successive amounts under compound interest directly reveals the yearly growth factor.

Final Answer:
The required annual compound interest rate is 5% per annum.