CAT 2022 Slot 2 · Quant · Question 18

The three slices must first be worked out as fractions of the capital.

One-fifth sits at 6% and one-third at 10%.

Together those account for \frac{1}{5}+\frac{1}{3}=\frac{3+5}{15}=\frac{8}{15}.

So the remainder is 1-\frac{8}{15}=\frac{7}{15}, invested at 1%.

With simple interest, the yearly income is the weighted sum of the rates.

That is \frac{1}{5}\times 6+\frac{1}{3}\times 10+\frac{7}{15}\times 1 percent of the capital.

Over a common denominator of 15, this is \frac{18}{15}+\frac{50}{15}+\frac{7}{15}=\frac{75}{15}.

So he earns exactly 5% of his capital each year.

For the accumulated interest to reach the capital itself, he needs 100%.

That takes \frac{100}{5}=20 years.

Shortcut. Simple interest means the rates just average out by weight, so the portfolio earns one blended rate forever. Recognising that blend as a round 5% makes the doubling time \frac{100}{5}=20 immediate.