For a certain principal invested at compound interest, the interest earned in the 3rd year is Rs. 12,100 at a fixed annual rate of 9% compounded yearly. What will be the compound interest (in rupees) earned in the 4th year on the same principal at the same rate?

Introduction / Context:
This question uses the relationship between successive yearly interests under compound interest. When a fixed rate is applied and interest is compounded annually, the interest for each year itself grows by the same factor (1 + r) from year to year. We are given the interest in the 3rd year and asked to find the interest in the 4th year.

Given Data / Assumptions:

Compound interest in 3rd year, I3 = Rs. 12,100
Annual rate of interest r = 9% per annum
Interest is compounded once per year
We must find compound interest in the 4th year, I4

Concept / Approach:
Under annual compounding at a constant rate r, each year's interest equals the previous year's interest multiplied by (1 + r/100). This is because the principal grows every year, and the new interest is computed on that increased principal. Therefore:
I4 = I3 * (1 + r/100). We simply multiply the 3rd year interest by (1.09) to get the 4th year interest.

Step-by-Step Solution:
Step 1: Identify the growth factor for interest. 1 + r/100 = 1 + 9/100 = 1.09. Step 2: Use the relationship between successive year interests. I4 = I3 * 1.09. I4 = 12,100 * 1.09. I4 = 12,100 + (12,100 * 0.09) = 12,100 + 1,089 = 13,189.

Verification / Alternative check:
We can cross-check by observing that if the principal at the start of the 3rd year is some value P3, then:
I3 = P3 * 0.09 = 12,100 ⇒ P3 = 12,100 / 0.09. At the start of the 4th year, the principal becomes P4 = P3 + I3 = P3 * 1.09. The interest in the 4th year would be P4 * 0.09 = P3 * 1.09 * 0.09 = I3 * 1.09, which matches our earlier formula and result. This confirms that multiplying by 1.09 is correct.

Why Other Options Are Wrong:
Values like Rs. 14,376 or Rs. 15,669 would correspond to much higher effective growth than 9% between the 3rd and 4th years. Rs. 17,080 is even more unrealistic. Only Rs. 13,189 maintains the correct 9% growth from the 3rd-year interest of Rs. 12,100.

Common Pitfalls:
Some candidates mistakenly add a flat 9% of the original principal rather than 9% of the increased principal, thereby ignoring the compounding effect. Others attempt to reconstruct the entire principal from scratch, which is unnecessary here. Recognizing that yearly interests under compound interest form a geometric progression can save considerable time.

Final Answer:
The compound interest in the 4th year will be Rs. 13,189.