Arithmetic

Simple & Compound Interest

Easy

Mr. Pinto invests one-fifth of his capital at $6%$ , one-third at $10%$ and the remaining at $1%$ , each rate being simple interest per annum. Then, the minimum number of years required for the cumulative interest income from these investments to equal or exceed his initial capital is

Entered answer:

✅ Correct Answer: 20

Mr. Pinto splits his money into three investments with different interest rates. We need to find when the total interest earned becomes equal to his original capital.

This is a classic compound planning problem that tests our understanding of simple interest and fraction handling.

The fractions in this problem are $\frac{1}{5}$ and $\frac{1}{3}$ . To avoid messy decimal calculations, we choose a number that's divisible by both 5 and 3.

LCM of 5 and 3 = 15

So let's assume total capital = 15P

This is a common trick in percentage and fraction problems - choose numbers that make calculations clean!

Now let's calculate each investment amount:

Investment 1: $\frac{1}{5}$ of capital at 6% SI

Amount = $\frac{1}{5} \times 15P = 3P$ at 6%

Investment 2: $\frac{1}{3}$ of capital at 10% SI

Amount = $\frac{1}{3} \times 15P = 5P$ at 10%

Investment 3: Remaining at 1% SI

Amount = $15P - (3P + 5P) = 7P$ at 1%

Quick check: $3P + 5P + 7P = 15P$

Simple Interest Formula: $SI = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100}$

For each year ( $t = 1$ ), the interest from each investment:

From 3P at 6%: $\frac{3P \times 6 \times 1}{100} = 0.18P$ per year

From 5P at 10%: $\frac{5P \times 10 \times 1}{100} = 0.50P$ per year

From 7P at 1%: $\frac{7P \times 1 \times 1}{100} = 0.07P$ per year

Total interest per year = $0.18P + 0.50P + 0.07P = 0.75P$

We need: Total Interest ≥ Initial Capital

For $t$ years: Total Interest = $0.75P \times t$

$0.75P \times t \geq 15P$

$0.75t \geq 15$

$t \geq \frac{15}{0.75} = 20$

Answer: 20 years

Key Learning: When dealing with multiple investments, we calculate the combined annual return first, then find when this total return equals our target amount.

Pro Tip: Always verify our fraction calculations by checking if all parts add up to the whole ( $3P + 5P + 7P = 15P$ in our case).

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Mr. Pinto invests one-fifth of his capital at $6%$ , one-third at $10%$ and the remaining at $1%$ , each rate being simple interest per annum. Then, the minimum number of years required for the cumulative interest income from these investments to equal or exceed his initial capital is