Simple Interest and Compound Interest Question 2 - CAT 2023
Anil invests Rs. 22,000 for 6 years in a certain scheme with 4% interest per annum, compounded half-yearly. Sunil invests in the same scheme for 5 years, and then reinvests the entire amount received at the end of 5 years for one year at 10% simple interest. If the amounts received by both at the end of 6 years are the same, then the initial investment made by Sunil, in rupees, is:
Step-by-Step Explanation
Let Sunil’s initial investment be S.
The scheme offers a 4% annual interest rate compounded half-yearly.
- Compounding rate per half-year = 4% / 2 = 2% (multiplier = 1.02).
- Anil's Final Position (6 years):
Six years contain 6 × 2 = 12 half-yearly intervals.
Amount = 22,000 × (1.02)¹²
- Sunil's Final Position (5 years compounding + 1 year simple interest):
Sunil accumulates money for 5 years (5 × 2 = 10 half-yearly intervals):
Maturity after 5 years = S × (1.02)¹⁰
He then moves this entire maturity value into a 10% simple interest scheme for 1 year:
Amount = [S × (1.02)¹⁰] × (1 + 0.10) = S × (1.02)¹⁰ × 1.1
- Equate both balances to isolate S:
S × (1.02)¹⁰ × 1.1 = 22,000 × (1.02)¹²
S × 1.1 = 22,000 × (1.02)²
S = 22,000 / 1.1 × 1.0404
S = 20,000 × 1.0404 = 20,808
Sunil's initial investment was Rs. 20,808.
Correct Answer: 20,808
💡 FundaMakers Tip:
Never panic when you see large exponents like (1.02)¹² on a non-calculator exam. Write down the complete horizontal expression for both sides before performing any scratchpad math. The exponents will almost always cancel each other out, leaving behind a small index difference like (1.02)², which can be expanded mentally.