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's assume Mr. Pinto's initial capital is C dollars.
He invests one-fifth of his capital at $6\%$,which means he invests $(\frac{1}{5})\times{C}$ dollars at $6\%$ interest per annum.The interest earned from this investment after t years is $(\frac{1}{5})\times{C}\times0.06\times{t}$.
He also invests one-third of his capital at $10\%$,which means he invests $(\frac{1}{3})\times{C}$ dollars at $10\%$ interest per annum.The interest earned from this investment after t years is $(\frac{1}{3})\times{C}\times0.10\times{t}$.
The remaining amount,which is $(1-\frac{1}{5}-\frac{1}{3})\times{C}=(11/15)\times{C}$, is invested at $1\%$ interest per annum.The interest earned from this investment after t years is $(\frac{11}{15})\times{C}\times0.01\times{t}$.
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:
$(\frac{1}{5})\times{C}\times0.06\times{t}+(1/3)\times{C}\times0.10\times{t}+(\frac{11}{15})\times{C}\times0.01\times{t≥C}$
Now,let's solve for t:
$(\frac{1}{5})\times0.06\times{t}+(\frac{1}{3})\times0.10\times{t}+(\frac{11}{15})\times0.01\times{t≥1}$
Simplify:
0.012t+0.0333t+0.0073t ≥ 1
Combine the terms:
0.0523t ≥ 1
Now, divide both sides by 0.0523:
$t ≥ \frac{1}{0.0523}$
t ≥ 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.