Question

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

Answer 1

Correct Answer: 20

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

He invests one-fifth of his capital at \( 6\% \), which means he invests \( \left( \frac{1}{5} \right) \times C \) dollars at \( 6\% \) interest per annum. The interest earned from this investment after \( t \) years is:

\(\left(\frac{1}{5}\right) \times C \times 0.06 \times t\) 

He also invests one-third of his capital at \( 10\% \), which means he invests \( \left( \frac{1}{3} \right) \times C \) dollars at \( 10\% \) interest per annum. The interest earned from this investment after \( t \) years is:

\(\left(\frac{1}{3}\right) \times C \times 0.10 \times t\)

The remaining amount, which is \( \left( 1 - \frac{1}{5} - \frac{1}{3} \right) \times C = \left( \frac{11}{15} \right) \times C \), is invested at \( 1\% \) interest per annum. The interest earned from this investment after \( t \) years is:

\(\left(\frac{11}{15}\right) \times C \times 0.01 \times t\)

Now, we want the cumulative interest income from these investments to equal or exceed his initial capital, which is \( C \) dollars. So, we can set up the following inequality:

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

Now, let's solve for \( t \):

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

Simplify:

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

Combine the terms:

\(0.0523t \geq 1\)

Now, divide both sides by 0.0523:

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

\(t \geq 19.13\)

Since the time (\( t \)) must be a whole number of years, the minimum number of years required for the cumulative interest income from these investments to equal or exceed his initial capital is 20 years.

Conclusion

So, the correct answer is 20 years.

Answer 2

Let the number of years needed be \( T \) years, and let the total investment be \( 15x \). 

The interest earned on the investments can be expressed as:

\(\frac{{3x \times 6 \times T}}{{100}} + \frac{{5x \times 10 \times T}}{{100}} + \frac{{7x \times 1 \times T}}{{100}} \geq 15x\)

We simplify this to:

\(\frac{75xT}{100} \geq 15x\)

And further simplify:

\(T \geq 20\)

Thus, 20 years is the minimal value of \( T \).