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

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

• He invests \( \frac{1}{5}C \) at \( 6\% \) interest
• He invests \( \frac{1}{3}C \) at \( 10\% \) interest
• The remaining \( C - \left( \frac{1}{5}C + \frac{1}{3}C \right) = \frac{11}{15}C \) is invested at \( 1\% \) interest

 Interest Accrued After \( t \) Years

Total interest after \( t \) years is:

\[ \text{Interest} = \left( \frac{1}{5}C \cdot 0.06 \cdot t \right) + \left( \frac{1}{3}C \cdot 0.10 \cdot t \right) + \left( \frac{11}{15}C \cdot 0.01 \cdot t \right) \]

This should be at least equal to the initial capital \( C \):

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

 Simplifying the Inequality

Compute each term: \[ \frac{1}{5} \cdot 0.06 = 0.012,\quad \frac{1}{3} \cdot 0.10 = 0.0333,\quad \frac{11}{15} \cdot 0.01 \approx 0.0073 \]

Adding them up: \[ 0.012 + 0.0333 + 0.0073 = 0.0526 \]

So the inequality becomes: \[ 0.0526t \geq 1 \Rightarrow t \geq \frac{1}{0.0526} \approx 19.01 \]

 Final Answer

Since \( t \) must be a whole number, the minimum number of years is: \[ \boxed{20 \text{ years}} \]