CAT 2023 Slot 1 · Quant · Question 15

Let the daily amounts be a, s and k for Amal, Sunil and Kamal.

Kamal takes twice as long as Amal, so he does half as much per day: k=\frac{a}{2}.

Quantities in harmonic progression have reciprocals in arithmetic progression, so \frac{1}{a}, \frac{1}{s}, \frac{1}{k} are in AP.

That means \frac{2}{s}=\frac{1}{a}+\frac{1}{k}.

Since \frac{1}{k}=\frac{2}{a}, this gives \frac{2}{s}=\frac{1}{a}+\frac{2}{a}=\frac{3}{a}.

So s=\frac{2a}{3}.

The job is completed by 4a+9s+16k=1.

Substituting s=\frac{2a}{3} and k=\frac{a}{2}: 4a+9\times \frac{2a}{3}+16\times \frac{a}{2}=1.

That is 4a+6a+8a=18a=1, so a=\frac{1}{18}.

Then s=\frac{2}{3}\times \frac{1}{18}=\frac{1}{27}.

Working alone, Sunil needs \frac{1}{s}=27 days.

Shortcut. In a harmonic progression the times are in arithmetic progression. Amal's time is t and Kamal's is 2t, so Sunil's is the middle term \frac{3t}{2}. Rates then stand as \frac{1}{t}, \frac{2}{3t}, \frac{1}{2t}, and the work equation gives t=18, so Sunil takes \frac{3}{2}\times 18=27 days.