Amal invests Rs $12000$ at 8% interest, compounded annually, and Rs $10000$ at 6% interest, compounded semi-annually, both investments being for one year. Bimal invests his money at $7.5 \%$ simple interest for one year. If Amal and Bimal get the same amount of interest, then the amount, in Rupees, invested by Bimal is

Entered answer:

✅ Correct Answer: 20920

We work through this, calculating each person's interest carefully.

Calculate Amal's Interest from Both Investments

Investment 1: Rs 12000 at 8% compounded annually

For compound interest compounded annually: Amount = Principal × (1 + Rate/100)^Time

Amount after 1 year = 12000 × (1 + 8/100)¹ = 12000 × 1.08 = Rs 12960

Interest earned = 12960 - 12000 = Rs 960

Investment 2: Rs 10000 at 6% compounded semi-annually

Semi-annually means the interest is compounded twice per year (every 6 months).

We adjust our formula:

Rate per period = 6% ÷ 2 = 3% per 6 months

Number of periods = 1 year × 2 = 2 periods

Amount = 10000 × (1 + 3/100)² = 10000 × (1.03)²

Calculate (1.03)²:

(1.03)² = 1.03 × 1.03 = 1.0609

Amount = 10000 × 1.0609 = Rs 10609

Interest earned = 10609 - 10000 = Rs 609

Total interest earned by Amal = 960 + 609 = Rs 1569

Set Up Equation for Bimal's Simple Interest

For simple interest: Interest = Principal × Rate × Time

Let Bimal's investment = Rs $x$

Bimal's interest = $x × \tfrac{7.5}{100} × 1 = 0.075x$

Since both earn the same interest:

Amal's interest = Bimal's interest

1569 = 0.075x

$x = \tfrac{1569}{0.075}$

To divide by 0.075, we multiply by $\tfrac{1000}{75} = \tfrac{40}{3}$

$x = 1569 × \tfrac{40}{3} = \tfrac{62760}{3} = 20920$

Bimal invested Rs 20920

Key insight: Semi-annual compounding means compounding twice per year, requiring us to convert the rate and time periods accordingly. The compound interest from multiple investments can be compared directly to simple interest when the total returns are equal.

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