If the amounts received at the end of the second and third year on a certain principal at compound interest are Rs. 1800 and Rs. 1926 respectively, what is the annual rate of interest?

Introduction / Context:
This problem checks your understanding of how consecutive amounts at compound interest are related by the interest rate. When you know the amount at the end of successive years, you can easily find the rate without first finding the principal. This is a common type of question in bank exams and aptitude tests involving compound interest.

Given Data / Assumptions:

• Amount at the end of 2nd year, A2 = Rs. 1800
• Amount at the end of 3rd year, A3 = Rs. 1926
• Interest is compounded annually
• Principal is the same for both years
• We need the annual rate of interest r (in percent)

Concept / Approach:
Under compound interest with annual compounding, each year the amount multiplies by the factor (1 + r / 100). Therefore, the ratio of the amount at the end of the third year to the amount at the end of the second year is:
A3 / A2 = 1 + r / 100From this ratio, we can directly solve for r without computing the principal.

Step-by-Step Solution:
Step 1: Write the relationship between 2nd and 3rd year amounts.A3 = A2 * (1 + r / 100)Step 2: Substitute the given values A3 = 1926 and A2 = 1800.1926 = 1800 * (1 + r / 100)Step 3: Divide both sides by 1800.1926 / 1800 = 1 + r / 1001926 / 1800 = 1.07Therefore, 1 + r / 100 = 1.07Step 4: Solve for r.r / 100 = 1.07 - 1 = 0.07r = 0.07 * 100 = 7%

Verification / Alternative check:
As a quick verification, assume an unknown principal P. After 2 years: A2 = P * (1.07)^2. After 3 years: A3 = P * (1.07)^3. Then A3 / A2 = (P * 1.07^3) / (P * 1.07^2) = 1.07, which matches 1926 / 1800 = 1.07. So the rate is indeed 7% per annum.

Why Other Options Are Wrong:
7.5% and 6.5%: These correspond to slightly larger or smaller ratios than 1.07 and do not match 1926 / 1800 exactly.6%: This would give 1.06 as the ratio, which is less than 1926 / 1800.8%: This would give 1.08, larger than the correct ratio.

Common Pitfalls:
Many learners try to find the principal first, which makes the calculation longer. Others mistakenly treat the difference between the amounts (1926 - 1800 = 126) as simple interest, which is incorrect in compound interest problems. The correct method is to use the ratio of consecutive amounts to directly obtain the multiplier and then the rate.

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