The compound interest on a certain principal for 2 years at 10% per annum is Rs 1155. What is the simple interest on the same principal for double this time, that is 4 years, at half the rate, that is 5% per annum?

Introduction / Context:
This question links compound interest and simple interest for the same principal but under different rates and time periods. First, we are given compound interest for 2 years at 10% and asked to infer the principal. Then, using that principal, we must compute simple interest for 4 years at 5%. This tests your ability to move between compound and simple interest frameworks and to reuse the principal in a second calculation.

Given Data / Assumptions:

• Compound interest CI for 2 years at 10% per annum is Rs 1155
• For the compound interest stage, time n1 = 2 years, rate r1 = 10% per annum
• For the simple interest stage, time n2 = 4 years, rate r2 = 5% per annum
• The principal P is the same in both stages
• We must find simple interest SI for the second stage

Concept / Approach:
For annual compounding, CI over n years is CI = P * [(1 + r / 100)^n - 1]. Here, with n1 = 2 years and r1 = 10%, we have CI = P * [(1.10)^2 - 1] = P * (1.21 - 1) = P * 0.21. From CI = 1155, we can solve for P. Once P is known, we apply the simple interest formula SI = P * r2 * n2 / 100 for 4 years at 5%. The process highlights how compound interest and simple interest formulas coexist and how to transition between them.

Step-by-Step Solution:
Stage 1: compound interest given CI = 1155, r1 = 10%, n1 = 2 years CI = P * [(1 + 10 / 100)^2 - 1] = P * (1.21 - 1) = P * 0.21 So 1155 = P * 0.21 P = 1155 / 0.21 = 5500 Stage 2: simple interest on the same P SI = P * r2 * n2 / 100 with P = 5500, r2 = 5%, n2 = 4 years SI = 5500 * 5 * 4 / 100 SI = 5500 * 20 / 100 = 5500 * 0.20 = 1100

Verification / Alternative check:
We can verify the principal calculation by recomputing the compound interest at 10% for 2 years. Amount A at the end of 2 years is A = 5500 * (1.10)^2 = 5500 * 1.21 = 6655. Compound interest is A - P = 6655 - 5500 = 1155, which matches the given value. For the simple interest stage, a 5% rate over 4 years gives a total percentage of 20%, and 20% of 5500 is 1100, again confirming the result.

Why Other Options Are Wrong:
5500 is actually the principal, not the required simple interest. 1400 and 4120 correspond to much larger interest amounts that would require either a higher rate, a longer time, or a different principal. They do not match the simple calculation with a 5% rate over 4 years on 5500. Only 1100 agrees with the formula and with the relationship between compound and simple interest across the two stages.

Common Pitfalls:
Some learners confuse the given CI with the principal itself or incorrectly apply the simple interest formula to the compound interest data without first finding P. Others forget to adjust the rate and time correctly in the second stage. Keeping the two stages of the problem clearly separated and labeling the rates and periods helps avoid such confusion.

Final Answer:
The simple interest on the same sum for 4 years at 5 percent per annum is Rs 1100.