CAT 2022 Slot 3 · Quant · Question 16

Work done is proportional to people multiplied by hours, so count person-hours in each phase.

In the first phase, N people work 7 hours a day for 10 days, that is 70 hours each.

So the first phase uses 70N person-hours and completes 35% of the project.

In the second phase, N-10 people work 10 hours a day for 14 days, that is 140 hours each.

So the second phase uses 140\left( N-10 \right) person-hours and completes the remaining 65%.

Person-hours are proportional to work done, so \frac{70N}{140\left( N-10 \right)}=\frac{35}{65}.

The right side simplifies to \frac{7}{13}.

Cross-multiplying, 70N\times 13=7\times 140\left( N-10 \right).

That is 910N=980N-9800.

Rearranging, 70N=9800, so N=140 — the second option.

Shortcut. The work ratio 35:65 is 7:13 and the hours ratio is 70:140, that is 1:2. So the headcount ratio must be \frac{7}{13}\times 2=\frac{14}{13}, giving \frac{N}{N-10}=\frac{14}{13} and N=140 directly.