Comparing Simple Interest in the Third Year with Compound Interest: On a certain principal, the simple interest earned in the 3rd year alone is ₹1,750 (so the yearly simple interest is constant). If the compound interest on the same sum for 2 years at the same rate (compounded annually) is ₹3,622.5, what is the annual rate of interest (in %)?

Introduction / Context:
This question links the simple interest earned in a single year (the 3rd year) with the total compound interest for 2 years on the same principal at the same rate. It tests your understanding that in simple interest the yearly interest is constant, while compound interest grows because interest itself earns interest.

Given Data / Assumptions:

• Principal = P.
• Simple interest in the 3rd year alone = ₹1,750.
• This implies yearly SI is constant at ₹1,750.
• Compound interest for 2 years at the same rate (annual compounding) = ₹3,622.5.
• Rate of interest per annum = r% (same for both SI and CI).

Concept / Approach:
For simple interest, interest per year is P * r / 100 and is the same in every year. So the 3rd year SI equals P * r / 100. For compound interest with annual compounding for 2 years, CI = P * ((1 + r / 100)^2 - 1). From the first relation, we can express P in terms of r and the known yearly SI of ₹1,750. Then we substitute this P into the CI formula and test option values of r until the computed CI equals ₹3,622.5.

Step-by-Step Solution:
Step 1: From SI, yearly interest = P * r / 100 = ₹1,750. Step 2: Rearranging, P = (1,750 * 100) / r. Step 3: For CI over 2 years: CI = P * ((1 + r / 100)^2 - 1) = ₹3,622.5. Step 4: Substitute P into the CI formula and test the options for r. Step 5: Try r = 7%. Then P = (1,750 * 100) / 7 = ₹25,000. Step 6: Compute CI at r = 7%: A = 25,000 * (1.07)^2 = 25,000 * 1.1449 = ₹28,622.5. Step 7: CI = A - P = 28,622.5 - 25,000 = ₹3,622.5, which exactly matches the given CI. Step 8: Therefore, the annual rate of interest is 7% per annum.

Verification / Alternative check:
With r = 7% and P = ₹25,000, the yearly SI is (25,000 * 7) / 100 = ₹1,750, confirming the 3rd year simple interest. The calculated CI for 2 years also matches the given ₹3,622.5. Since both conditions are satisfied for r = 7% and no other option will satisfy both simultaneously, the answer is confirmed.

Why Other Options Are Wrong:

• 8%: Gives a different principal and CI around ₹3,640, not ₹3,622.5.
• 9%: Produces CI that is larger than required, around ₹3,657.5.
• 10%: Leads to an even higher CI, not matching the given value.
• 6%: Results in CI around ₹3,605, which is too low.

Common Pitfalls:
Some students mistakenly treat the 3rd year simple interest as 3 times the annual interest or mix up SI and CI formulas. Remember that in simple interest, yearly interest is constant, so the 3rd year SI is exactly equal to the yearly SI. Always set up the equations carefully before substituting values from the options.

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