On a certain principal at compound interest, the amount at the end of the 2nd year is Rs. 1,800 and the amount at the end of the 3rd year is Rs. 1,926. What is the annual rate of interest?

Introduction / Context:
This question uses amounts at the end of successive years under compound interest to determine the annual rate. It is a useful technique because the ratio of amounts in consecutive years directly reveals the growth factor 1 + r, and hence the annual rate r, without needing the original principal.

Given Data / Assumptions:

• Amount at the end of year 2, A2 = Rs. 1,800.
• Amount at the end of year 3, A3 = Rs. 1,926.
• Interest is compounded annually at a constant rate r.
• We must find the annual rate of interest r in percent.

Concept / Approach:
Under annual compounding, the amount each year is multiplied by the factor (1 + r). Therefore A3 = A2 * (1 + r). This means the ratio A3 / A2 equals (1 + r). Once we compute the ratio, we subtract 1 to find r. This method saves us from solving for the principal and is often faster and less error prone.

Step-by-Step Solution:
Step 1: Use the relationship A3 = A2 * (1 + r).Step 2: Rearrange to get 1 + r = A3 / A2.Step 3: Substitute the given values: 1 + r = 1,926 / 1,800.Step 4: Compute the ratio 1,926 / 1,800 = 1.07.Step 5: Therefore, 1 + r = 1.07, which gives r = 0.07.Step 6: Convert r to percent: 0.07 corresponds to 7% per annum.

Verification / Alternative check:
We can verify by checking the difference between the two amounts. The increase from year 2 to year 3 is 1,926 - 1,800 = Rs. 126. This increase should equal r times 1,800 if the rate is constant. At 7%, the interest on 1,800 is 0.07 * 1,800 = Rs. 126, which matches the actual increase. This confirms that the annual rate is indeed 7%.

Why Other Options Are Wrong:

• 7.5%: At 7.5%, the factor 1 + r would be 1.075, giving A3 = 1,800 * 1.075 = 1,935, not 1,926.
• 6%: At 6%, the factor is 1.06, leading to A3 = 1,800 * 1.06 = 1,908, which is too low.
• 6.5%: At 6.5%, the factor is 1.065, giving A3 = 1,800 * 1.065 = 1,917, still less than 1,926.

Common Pitfalls:
Some learners mistakenly treat the amounts as simple interest and attempt to subtract an equal interest amount each year. Others try to solve for the principal unnecessarily, making the calculations more complicated. It is also easy to make mistakes in division, so careful calculation of A3 / A2 is important. Remember that for compound interest, consecutive amounts have a simple multiplicative relationship, which makes this type of question straightforward when recognized.

Final Answer:
The annual rate of compound interest is 7% per annum.