CAT 2024 Slot 2 · Quant · Question 6

Let a, v and s be the daily work rates of Amal, Vimal and Sunil, with the whole task taken as 1.

The pair rates give a+v=\frac{1}{150} and v+s=\frac{1}{100}.

The relay in the question gives 75a+135v+45s=1.

From the first pair relation, 75a+75v=\frac{75}{150}=\frac{1}{2}.

From the second, 45v+45s=\frac{45}{100}=\frac{9}{20}.

Adding these two, 75a+120v+45s=\frac{1}{2}+\frac{9}{20}=\frac{19}{20}.

Subtracting this from the relay equation leaves 15v=1-\frac{19}{20}=\frac{1}{20}, so v=\frac{1}{300}.

Then a=\frac{1}{150}-\frac{1}{300}=\frac{1}{300} and s=\frac{1}{100}-\frac{1}{300}=\frac{2}{300}.

Take the task as 300 units, so Amal does 1 unit a day, Vimal 1 unit a day and Sunil 2 units a day.

Under the new arrangement Amal works daily, Vimal on days 2,4,6,\ldots and Sunil on days 3,6,9,\ldots, so the pattern repeats every 6 days.

Within one such block the daily outputs are 1,2,3,2,1,4, totalling 13 units per 6 days.

Now 300=13\times 23+1, so 23 full blocks give 299 units in 23\times 6=138 days.

Only 1 unit is left, and day 139 is an Amal-only day contributing exactly 1 unit.

The task is finished on day 139.

Shortcut. Scaling the task to the LCM-friendly 300 units turns every rate into a whole number, and the cycle total of 13 units is then read straight off the six-day grid — no fractions survive past that point.