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

Introduction / Context:
This question connects simple interest and compound interest. First, we are told that a certain principal grows by 50% in 5 years at simple interest, which allows us to deduce the annual rate of interest. Then we must apply this same rate to a new principal of Rs 12,000, but now under compound interest for 3 years. The problem thus tests understanding of both simple interest and compound interest formulas and how to move from one to the other using the underlying rate.

Given Data / Assumptions:

• Under simple interest, an amount increases by 50% in 5 years.
• This means simple interest over 5 years is 50% of the principal.
• We infer the annual simple interest rate from this information.
• For the second part, principal P2 = Rs 12,000.
• Rate R is the same as the rate found from the simple interest condition.
• Time T2 = 3 years, and interest is compounded annually.

Concept / Approach:
For simple interest, the interest is:
SI = P * R * T / 100If in 5 years the amount increases by 50%, then SI over 5 years is 50% of P, so:
P * R * 5 / 100 = 0.5 * PFrom this we can solve for R. Once R is known, the compound interest on Rs 12,000 for 3 years is found using:
A = 12000 * (1 + R / 100)^3CI = A - 12000

Step-by-Step Solution:
Step 1: From the simple interest condition, P cancels. We have R * 5 / 100 = 0.5.Step 2: Multiply both sides by 100: 5R = 50, so R = 10% per annum.Step 3: Now apply this rate to Rs 12,000 with compound interest for 3 years. Amount A = 12000 * (1 + 10 / 100)^3.Step 4: Compute 1 + 10 / 100 = 1.10.Step 5: Compute 1.10^3 = 1.10 * 1.10 * 1.10 = 1.331.Step 6: Multiply principal by this factor: A = 12000 * 1.331 = 15972.Step 7: Compound interest CI = A - P = 15972 - 12000 = 3972.

Verification / Alternative check:
An alternative way to see this is to build the amount year by year. Year 1: amount = 12000 + 10% of 12000 = 12000 + 1200 = 13200. Year 2: amount = 13200 + 10% of 13200 = 13200 + 1320 = 14520. Year 3: amount = 14520 + 10% of 14520 = 14520 + 1452 = 15972. Again, CI = 15972 - 12000 = 3972. This matches the previous calculation and confirms the answer.

Why Other Options Are Wrong:
Rs.5422, Rs.5489, and Rs.6145 are significantly larger than Rs 3,972 and would require a higher effective rate or longer time. Rs.3600 corresponds to 10% simple interest for 3 years on 12,000 (which would be 12,000 * 10 * 3 / 100 = 3600), but the question specifically asks for compound interest. Therefore, Rs.3600 is a common but incorrect simple interest based answer. Only Rs.3972 correctly reflects compound interest at 10% per annum for 3 years.

Common Pitfalls:
Students often compute the rate correctly but then revert to simple interest when calculating interest on Rs 12,000, or they mistakenly apply the 50% increase directly to 3 years instead of determining the correct annual rate. Another frequent mistake is to miscalculate 1.10^3, writing it as 1.30 or 1.33 incorrectly. Writing out each multiplication step or using a year by year method is a good way to avoid such errors.

Final Answer:
The compound interest on Rs 12,000 after 3 years at the same rate that produces a 50% increase in 5 years under simple interest is Rs.3972.