Compound interest earned on a sum for the second and third years is Rs. 1,200 and Rs. 1,440 respectively. Find the annual rate of interest (per annum). Assume yearly compounding.

Introduction / Context:
This question uses the relationship between compound interest amounts in successive years to find the rate of interest. When interest is compounded annually at a constant percentage, the interest earned each year forms a geometric progression, with each year's interest equal to the previous year's interest multiplied by (1 + r).

Given Data / Assumptions:

Interest earned in the 2nd year = Rs. 1,200.
Interest earned in the 3rd year = Rs. 1,440.
Compounding is annual at a constant rate r% per annum.
We are asked to find r.

Concept / Approach:
Under annual compounding, interest for successive years increases by a factor of (1 + r/100) each year, because each new year's interest is computed on an increased principal (previous principal plus previous interest). Thus: Interest in 3rd year = Interest in 2nd year * (1 + r/100). Taking the ratio of third year interest to second year interest directly reveals (1 + r/100).

Step-by-Step Solution:
Step 1: Set up the ratio of third year to second year interest. Interest in 2nd year = 1,200. Interest in 3rd year = 1,440. Ratio = 1,440 / 1,200 = 1.2. Step 2: Relate this ratio to the rate. Interest in 3rd year = Interest in 2nd year * (1 + r/100). So, 1 + r/100 = 1.2. Step 3: Solve for r. 1 + r/100 = 1.2 → r/100 = 0.2 → r = 20%.

Verification / Alternative check:
If the second year interest is 1,200 at 20%, the effective principal at the start of year 2 must be 1,200 / 0.20 = 6,000. At the start of year 3, principal becomes 6,000 + 1,200 = 7,200. Interest in year 3 at 20% is 7,200 * 0.20 = 1,440, which matches the given value, confirming r = 20%.

Why Other Options Are Wrong:
At 15%, the ratio of successive year interests would be 1.15; at 18% it would be 1.18; at 24% it would be 1.24. None of these match the observed ratio of 1.2 between 1,440 and 1,200. Therefore, only 20% is consistent with the data.

Common Pitfalls:
Some students try to involve the principal and write multiple equations unnecessarily. Others confuse the difference between interests for different years with the total interest over multiple years. The simplest method is to use the ratio of successive year interests directly to find the rate.

Final Answer:
The annual rate of interest is 20% per annum.