Introduction / Context:
This question first provides information about growth under simple interest and then asks you to apply the same annual rate to a different principal under compound interest. You must begin by extracting the underlying simple interest rate from the long-term increase, and then use that rate to compute a compound interest amount over a shorter period.
Given Data / Assumptions:
- Under simple interest, an amount increases by 40% in 8 years.
- We must find the equivalent annual rate of interest r%.
- Then, for a principal of Rs 30,000, we must calculate the compound interest for 2 years at the same rate.
- Interest for the second part is compounded annually.
Concept / Approach:
For simple interest, an increase of 40% in 8 years means the total simple interest over 8 years is 40% of the principal. So the rate per year is (40% / 8). Once we know r, we can calculate compound interest on Rs 30,000 for 2 years using CI = P * [(1 + r/100)^2 - 1].
Step-by-Step Solution:
Step 1: Simple interest increase of 40% in 8 years means:
Total SI = 40% of principal over 8 years.
Step 2: Rate per year r = 40 / 8 = 5% per annum.
Step 3: Now compute compound interest on Rs 30,000 for 2 years at 5%:
CI = P * [(1 + r/100)^2 - 1].
Step 4: Compute the factor:
1 + r/100 = 1 + 5/100 = 1.05.
Step 5: Square the factor:
1.05^2 = 1.1025.
Step 6: CI = 30000 * (1.1025 - 1) = 30000 * 0.1025 = 3075.
Verification / Alternative check:
Compute the final amount:
A = 30000 * 1.1025 = 33075.
Check: CI = A - P = 33075 - 30000 = 3075. This matches the calculated value.
Why Other Options Are Wrong:
4612.5 and 6150 would require higher effective rates or longer time periods than given.
7687.5 is much too large and does not correspond to 5% per year for 2 years.
Common Pitfalls:
One frequent mistake is to treat the 40% as an annual rate rather than an 8-year total increase. Another is to compute simple interest again in the second part, instead of compound interest, which would yield 30000 * 5% * 2 = 3000, slightly different from the correct compound interest.
Final Answer:
The compound interest on Rs 30,000 after 2 years at the same rate is Rs 3075.