For a certain principal invested at compound interest, the interest earned in the 3rd year is Rs. 125 and the interest earned in the 4th year is Rs. 135. Assuming annual compounding, what is the annual rate of interest (in percent)?

Introduction / Context:
This problem focuses on successive yearly interests under compound interest. When interest is compounded annually at a constant rate, the interest earned each year forms a geometric progression. We are given the interest in the 3rd and 4th years and are asked to find the underlying rate of interest.

Given Data / Assumptions:

Interest in 3rd year, I3 = Rs. 125
Interest in 4th year, I4 = Rs. 135
Interest is compounded annually at a constant rate r%
Principal is fixed but not needed explicitly
We must find r (the annual rate in percent)

Concept / Approach:
Under annual compounding at rate r%, the principal grows each year, so the interest for each subsequent year is obtained by multiplying the previous year's interest by (1 + r/100). Therefore:
I4 = I3 * (1 + r/100). This simple relation allows us to find the rate directly using the ratio I4 / I3.

Step-by-Step Solution:
Step 1: Compute the ratio of successive interests. I4 / I3 = 135 / 125 = 1.08. Step 2: Relate this to the growth factor. I4 = I3 * (1 + r/100) ⇒ 1 + r/100 = 1.08. Step 3: Solve for r. r/100 = 1.08 − 1 = 0.08. Thus, r = 8% per annum.

Verification / Alternative check:
Suppose the principal at the start of the 3rd year is P3. Then I3 = P3 * 0.08 = 125 ⇒ P3 = 125 / 0.08 = 1562.50. At the start of the 4th year, the principal becomes P4 = P3 + I3 = 1562.5 + 125 = 1687.5. The interest in the 4th year at 8% is then 1687.5 * 0.08 = 135, which matches the given I4 and confirms that the rate is 8%.

Why Other Options Are Wrong:
At 9%, the ratio would be 1.09, not 1.08. At 10%, it would be 1.10, and at 12%, it would be 1.12. None of these ratios matches 135 / 125. Only an 8% rate produces the correct progression from 125 to 135.

Common Pitfalls:
A common misconception is to try to express I3 and I4 in terms of the original principal and then solve using more complicated algebra, when the simple ratio method is much quicker. Another mistake is to assume the difference between interests is a flat amount and attempt to use simple interest logic, ignoring that the base on which interest is calculated has itself increased each year under compounding.

Final Answer:
The annual rate of interest is 8%.