Question:medium

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

Updated On: Jan 15, 2026
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Correct Answer: 20

Solution and Explanation

Assume Mr. Pinto's initial capital is \( C \) dollars.

He invests \( \left( \frac{1}{5} \right) C \) at \( 6\% \) annual interest. The interest earned after \( t \) years is \( \left(\frac{1}{5}\right) \times C \times 0.06 \times t \).

He invests \( \left( \frac{1}{3} \right) C \) at \( 10\% \) annual interest. The interest earned after \( t \) years is \( \left(\frac{1}{3}\right) \times C \times 0.10 \times t \).

The remaining capital, \( \left( 1 - \frac{1}{5} - \frac{1}{3} \right) C = \left( \frac{11}{15} \right) C \), is invested at \( 1\% \) annual interest. The interest earned after \( t \) years is \( \left(\frac{11}{15}\right) \times C \times 0.01 \times t \).

The cumulative interest income from these investments must equal or exceed his initial capital, \( C \). This can be represented by the inequality:

\(\left(\frac{1}{5}\right) \times C \times 0.06 \times t + \left(\frac{1}{3}\right) \times C \times 0.10 \times t + \left(\frac{11}{15}\right) \times C \times 0.01 \times t \geq C\)

Solving for \( t \):

\(\left(\frac{1}{5}\right) \times 0.06 \times t + \left(\frac{1}{3}\right) \times 0.10 \times t + \left(\frac{11}{15}\right) \times 0.01 \times t \geq 1\)

Simplifying the equation:

\(0.012t + 0.0333t + 0.0073t \geq 1\)

Combining terms:

\(0.0523t \geq 1\)

Dividing by 0.0523:

\(t \geq \frac{1}{0.0523}\)

\(t \geq 19.13\)

Since time must be a whole number of years, the minimum duration for the cumulative interest to meet or exceed the initial capital is 20 years.

Conclusion

The required time is 20 years.

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