There is a 50% increase in an amount in 5 years at simple interest. Using the same annual rate, what will be the compound interest on Rs. 12,000 after 3 years, compounded annually?

Introduction / Context:
This question links simple interest and compound interest. First we deduce the annual simple interest rate from the given percentage increase over 5 years. Then we use that rate as a compound interest rate to compute the compound interest on a different principal over 3 years.

Given Data / Assumptions:

An amount increases by 50% in 5 years at simple interest.
We need the equivalent annual rate of interest.
Principal for the second part = Rs. 12,000.
Time for compound interest = 3 years.
Compounding is annual in the second part.

Concept / Approach:
For simple interest: Simple interest SI = P * r * t / 100. A 50% increase over 5 years means SI = 0.5P in 5 years. This allows us to find r. Then, for compound interest: Amount A = P * (1 + r/100)^t, CI = A − P. We then compute CI for P = 12,000 at the found rate for 3 years.

Step-by-Step Solution:
Step 1: Find the simple interest rate. In 5 years, SI = 50% of P = 0.5P. Using SI formula: 0.5P = P * r * 5 / 100. 0.5 = 5r / 100 → r = 0.5 * 100 / 5 = 10% per annum. Step 2: Use r = 10% as the compound interest rate. Principal for CI = Rs. 12,000, t = 3 years. Amount A = 12,000 * (1.10)^3. (1.10)^3 = 1.331, so A = 12,000 * 1.331 = Rs. 15,972. Step 3: Compute CI on Rs. 12,000. CI = A − P = 15,972 − 12,000 = Rs. 3,972.

Verification / Alternative check:
We can compute year by year. End of year 1: amount = 12,000 * 1.10 = 13,200. Year 2: amount = 13,200 * 1.10 = 14,520. Year 3: amount = 14,520 * 1.10 = 15,972. The interest earned over 3 years is 15,972 − 12,000 = 3,972, confirming our result.

Why Other Options Are Wrong:
The other values (5,422; 5,489; 6,145) represent much higher compound interest than is consistent with a 10% rate over 3 years on Rs. 12,000. They would imply different, incorrect interest rates or time periods.

Common Pitfalls:
Some learners misinterpret the 50% increase over 5 years and compute the annual rate wrongly, for example dividing 50 by 5 without relating it to the simple interest formula. Others forget that the next part is compound interest, not simple interest, and simply apply SI again. Carefully distinguishing between SI and CI is essential.

Final Answer:
The compound interest on Rs. 12,000 for 3 years at the same rate is Rs. 3972.