The compound interest on a certain sum for 2 years at 10% per annum, compounded annually, is Rs 525. What is the simple interest on the same sum for double the time (4 years) at half the rate, that is, at 5% per annum?

Introduction / Context:
This question connects a known compound interest situation with a new simple interest situation on the same principal. We are given the compound interest for 2 years at 10% per annum and must first find the principal. Then we calculate the simple interest for 4 years at half the original rate, that is, at 5% per annum. This tests the ability to move between compound and simple interest correctly.

Given Data / Assumptions:

• Compound interest CI for 2 years at 10% per annum = Rs 525.
• Rate for CI r1 = 10% per annum.
• Time for CI t1 = 2 years.
• Rate for SI r2 = 5% per annum.
• Time for SI t2 = 4 years.
• Interest is compounded annually in the CI part.

Concept / Approach:
For 2 years at compound interest, the amount is:
A = P * (1 + r1 / 100)^2 The compound interest CI is:
CI = A - P = P * [(1 + r1 / 100)^2 - 1] From CI we can determine P. Then, simple interest for 4 years at 5% is:
SI = P * r2 * t2 / 100

Step-by-Step Solution:
Given CI = 525, r1 = 10%, t1 = 2 years. For 2 years, amount factor = (1.10)^2 = 1.21. Compound interest factor = 1.21 - 1 = 0.21. So CI = 0.21 * P = 525. Therefore P = 525 / 0.21 = Rs 2500. Now compute SI for 4 years at 5%. SI = P * r2 * t2 / 100 = 2500 * 5 * 4 / 100. SI = 2500 * 20 / 100 = 2500 * 0.20 = Rs 500.

Verification / Alternative Check:
Check CI with P = 2500. Amount after 2 years: A = 2500 * 1.21 = 3025. CI = A - P = 3025 - 2500 = Rs 525, matching the given CI. This confirms P is correct and validates the simple interest calculation.

Why Other Options Are Wrong:
Rs. 4000 is far too large and would correspond to an unrealistically high rate or time. Rs. 600 and Rs. 800 are larger than the correct SI and come from incorrect use of rate or time.

Common Pitfalls:
Some learners confuse CI and SI formulas and may attempt to apply simple interest logic when finding P. Another error is miscomputing the amount factor (1.10)^2.

Final Answer:
The simple interest for 4 years at 5% per annum is Rs. 500.