CAT 2022 Slot 3 · Quant · Question 3

Bob alone takes 40 days, so his rate is \frac{1}{40} of the job per day.

Alex is twice as fast, so his rate is \frac{2}{40}=\frac{1}{20}.

Alex is thrice as fast as Cole, so Cole's rate is \frac{1}{20}\div 3=\frac{1}{60}.

Take the job as 120 units: Bob does 3 units a day, Alex 6 and Cole 2.

Day 1 is Alex and Bob, producing 6+3=9 units.

Day 2 is Bob and Cole, producing 3+2=5 units.

Day 3 is Cole and Alex, producing 2+6=8 units.

One full cycle of three days therefore delivers 9+5+8=22 units.

Five cycles cover 15 days and 5\times 22=110 units, leaving 10 units.

Day 16 repeats day 1 and adds 9 units, bringing the total to 119 with 1 unit left.

Day 17 repeats day 2 — Bob and Cole — and they finish that last unit, with Alex not working.

Alex works on the first and third day of every cycle, that is days 1, 3, 4, 6, 7, 9, 10, 12, 13 and 15, plus day 16.

Counting those, Alex worked on 11 days.

Shortcut. Alex works two days in every three, so after 15 days he has worked 10. The only question is how far into the sixth cycle the job runs — and since the leftover 10 units are cleared on days 16 and 17, Alex picks up exactly one more day.