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 is the same style of question as an earlier one, where we determine the annual compound interest rate using the amounts at the end of two consecutive years. It helps reinforce the idea that consecutive amounts under compound interest are related by a simple multiplication factor, which directly gives the rate.

Given Data / Assumptions:

• Amount after year 2, A2 = Rs. 1,800.
• Amount after year 3, A3 = Rs. 1,926.
• Interest is compounded annually at a constant rate r.
• We must find r in percent per annum.

Concept / Approach:
Under annual compounding, A3 = A2 * (1 + r). Therefore the ratio A3 / A2 equals 1 + r. This allows us to find r directly from the two amounts without involving the principal. This approach is efficient and reduces the possibility of error compared with methods that attempt to solve for both principal and rate.

Step-by-Step Solution:
Step 1: Use the relation A3 = A2 * (1 + r).Step 2: Rearrange to find 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: So 1 + r = 1.07, hence r = 0.07.Step 6: Convert r to percent: 0.07 = 7% per annum.

Verification / Alternative check:
Check the difference between A3 and A2. The increase is 1,926 - 1,800 = Rs. 126. At 7%, the interest on 1,800 is 0.07 * 1,800 = Rs. 126, which matches the difference exactly. This confirms that 7% per annum is the correct rate.

Why Other Options Are Wrong:

• 7.5%: This would imply a factor of 1.075 and A3 would be 1,800 * 1.075 = 1,935, not 1,926.
• 6%: A factor of 1.06 would give A3 = 1,908, which is less than 1,926.
• 6.5%: A factor of 1.065 gives A3 = 1,917, again less than the given amount.

Common Pitfalls:
Learners may mistakenly use the simple interest formula or attempt to average the amounts instead of using the ratio method. It is also easy to make errors in division or in translating the decimal 0.07 into 7%. To avoid mistakes, always compute the ratio carefully and remember that multiplying by 100 converts a decimal rate into a percentage.

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