The compound interest earned on a certain sum for the second year and the third year are Rs 1,200 and Rs 1,440 respectively. Assuming a constant annual rate with yearly compounding, what is the rate of interest per annum?

Introduction / Context:
This question involves analyzing compound interest year by year, instead of over the whole period. We are given the interest earned in the second and third years and asked to deduce the annual interest rate. Under compound interest, the interest in each successive year increases because it is calculated on a growing amount. The ratio of interest in consecutive years depends directly on the annual rate. This question tests understanding of that relationship without requiring the explicit principal.

Given Data / Assumptions:

• Compound interest in the second year = Rs 1,200.
• Compound interest in the third year = Rs 1,440.
• Interest is compounded annually.
• The rate of interest per annum is constant.
• Principal P is unknown and does not need to be found.

Concept / Approach:
Let the rate of interest be R% per annum and principal be P. Under annual compounding:
Amount after year 1 = P * (1 + R / 100).Amount after year 2 = P * (1 + R / 100)^2.Amount after year 3 = P * (1 + R / 100)^3.The interest earned in the second year is the difference between the amount at the end of year 2 and the amount at the end of year 1, and similarly for the third year. However, a simpler fact is that the interest in each successive year is multiplied by a factor of (1 + R / 100). Thus:
CI in third year = CI in second year * (1 + R / 100).

Step-by-Step Solution:
Step 1: Let I2 be the interest in the second year and I3 be the interest in the third year. We are given I2 = 1200 and I3 = 1440.Step 2: Under compound interest, we have I3 = I2 * (1 + R / 100).Step 3: Substitute the known values: 1440 = 1200 * (1 + R / 100).Step 4: Divide both sides by 1200 to get 1440 / 1200 = 1 + R / 100.Step 5: Simplify the fraction: 1440 / 1200 = 1.2.Step 6: Therefore, 1.2 = 1 + R / 100, so R / 100 = 0.2.Step 7: Multiply both sides by 100: R = 20.Step 8: So the rate of interest is 20% per annum.

Verification / Alternative check:
We can verify using a hypothetical principal P. If the rate is 20%, then the interest in year 2 is based on the amount at the end of year 1. Let amount at end of year 1 be A1. Then year 2 interest is A1 * 20 / 100 = 0.2 * A1. Year 3 interest is A1 * (1.2) * 0.2 = 0.24 * A1. The ratio of year 3 interest to year 2 interest is 0.24 * A1 / (0.2 * A1) = 1.2. Since the given ratio 1440 / 1200 also equals 1.2, the rate 20% fits perfectly.

Why Other Options Are Wrong:
At 15% per annum, the ratio of interest in successive years would be 1.15, not 1.2. At 18% per annum, the ratio would be 1.18, and at 24%, it would be 1.24. None of these match 1.2. A rate of 12% per annum gives a ratio of 1.12. Thus these options do not fit the observed ratio of third year interest to second year interest. Only 20% per annum produces exactly 1.2 as required by the given data.

Common Pitfalls:
Some students attempt to find the principal P unnecessarily and get bogged down in lengthy algebra, even though it cancels out. Others mistakenly use the formula for total compound interest instead of focusing on the year-wise interest amounts. A clear understanding that each year's interest under compound interest grows by the factor (1 + R / 100) simplifies the problem significantly.

Final Answer:
The rate of interest that makes the second year compound interest equal to Rs 1,200 and the third year compound interest equal to Rs 1,440 is 20% per annum.