On a certain principal, the compound interest received in the second year is Rs. 1200 at a rate of 10% per annum. What will be the compound interest for the fourth year on the same principal at the same rate?

Introduction / Context:
This question focuses on the concept of interest earned in specific years under compound interest. Instead of total interest over many years, we are given the interest in the second year and asked to find the interest in the fourth year. This requires recognizing the pattern of annual interest amounts under compound growth.

Given Data / Assumptions:

• Compound interest in 2nd year I2 = Rs. 1200
• Rate of interest r = 10% per annum
• We need compound interest in 4th year I4
• Principal P is the same in all years
• Compounding is annual

Concept / Approach:
Under compound interest, interest in a particular year grows because it is calculated on the amount at the beginning of that year. The interest in year n is:
In = P * (1 + r / 100)^(n - 1) * (r / 100)Thus:
I2 = P * (1 + r / 100)^(1) * (r / 100)I4 = P * (1 + r / 100)^(3) * (r / 100)Therefore:
I4 / I2 = (1 + r / 100)^2We can find I4 without explicitly computing P.

Step-by-Step Solution:
Step 1: Compute the growth factor.1 + r / 100 = 1 + 10 / 100 = 1.1Step 2: Use the ratio of interests.I4 / I2 = (1.1)^2(1.1)^2 = 1.21Step 3: Compute I4.I4 = I2 * 1.21I4 = 1200 * 1.21 = Rs. 1452

Verification / Alternative check:
If we wanted, we could find principal P. Let A1 = P * 1.1, interest in 2nd year is 10% of A1:
I2 = 0.1 * P * 1.1 = 0.11P = 1200P = 1200 / 0.11 = 10909.09 (approx)Amount at beginning of 4th year is P * (1.1)^3, so interest in 4th year is 10% of that amount:
I4 = 0.1 * P * (1.1)^3 ≈ 0.1 * 10909.09 * 1.331 ≈ 1452This confirms the earlier ratio method.

Why Other Options Are Wrong:
Rs. 1320 and Rs. 1420: These values are too low and do not reflect the correct growth factor of (1.1)^2.Rs. 1552 and Rs. 1600: These overestimate the increase from the second to the fourth year.

Common Pitfalls:
Some learners think that the interest per year stays the same or increases linearly, but under compound interest, it increases multiplicatively. Others try to calculate total amounts year by year and risk arithmetic errors. Using the ratio method based on (1 + r / 100)^2 is both faster and less error-prone.

Final Answer:
The compound interest for the fourth year on the same principal at 10% per annum is Rs. 1452.