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CAT 2022 Slot 3 — QA Question 19

Working AlternatelyEasy

Bob can finish a job in 40 days, if he works alone. Alex is twice as fast as Bob and thrice as fast as Cole in the same job. Suppose Alex and Bob work together on the first day, Bob and Cole work together on the second day, Cole and Alex work together on the third day, and then, they continue the work by repeating this three-day roster, with Alex and Bob working together on the fourth day, and so on. Then, the total number of days Alex would have worked when the job gets finished, is

Answer & solution

Answer: 11

Solution

Easy

Convert speeds to a convenient total work of 120120 units. Add up one 33-day cycle, count full cycles, then finish off the last few units day by day — tracking only the days Alex is on duty.

1

Times and efficiencies. Alex is twice Bob (2020 days) and thrice Cole (6060 days). Take total work =120=120 units (LCM):

Alex=12020=6,Bob=12040=3,Cole=12060=2 units/day\begin{aligned} &\text{Alex}=\frac{120}{20}=6,\quad \text{Bob}=\frac{120}{40}=3,\quad \text{Cole}=\frac{120}{60}=2\ \text{units/day} \end{aligned}
2

Work per 3-day cycle. Day1 Alex+Bob, Day2 Bob+Cole, Day3 Cole+Alex:

(6+3)⏟9+(3+2)⏟5+(2+6)⏟8=22 units\underbrace{(6+3)}_{9}+\underbrace{(3+2)}_{5}+\underbrace{(2+6)}_{8}=22\ \text{units}
3

Count complete cycles.

5 cycles =5×22=110 units in 15 daysremaining=120−110=10 units\begin{aligned} &5\text{ cycles }=5\times 22=110\ \text{units in }15\text{ days}\\ &\text{remaining}=120-110=10\ \text{units} \end{aligned}
4

Finish the last 10 units. Day 16 is Alex+Bob =9=9 units; 11 unit remains, finished Day 17 by Bob+Cole:

Day 16: 10−9=1 unit leftDay 17: Bob+Cole finish it — Alex not needed\begin{aligned} &\text{Day 16: }10-9=1\ \text{unit left}\\ &\text{Day 17: Bob+Cole finish it — Alex not needed} \end{aligned}
5

Count Alex's working days. In each cycle Alex works Day1 and Day3 = 22 days, plus Day 16:

5×2+1=11 days5\times 2+1=11\ \text{days}

Alex worked 11\mathbf{11} days.

CAT 2022 Slot 3 QA Q19: Bob can finish a job in 40 days, if he works alone. Alex is twice as fast as Bob and thrice as fast as Cole in — Solution | TheCATExam