Roster Work Days Calculation

CAT 2022 Slot 3 · QA · Medium · Arithmetic

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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

11

Explanation

Let the efficiencies of Alex, Bob, and Cole be A,B,CA, B, C. Bob takes 40 days, so let total work =120= 120 units.

  • Bob's efficiency B=3B = 3 units/day.
  • Alex is twice as fast as Bob   ⟹  A=6\implies A = 6 units/day.
  • Alex is thrice as fast as Cole   ⟹  C=2\implies C = 2 units/day.

Work done in a 3-day roster:

  • Day 1 (A + B): 6+3=96 + 3 = 9 units
  • Day 2 (B + C): 3+2=53 + 2 = 5 units
  • Day 3 (C + A): 2+6=82 + 6 = 8 units

Total work in one 3-day cycle =9+5+8=22= 9 + 5 + 8 = 22 units.

In 5 complete cycles (15 days): Work completed =5×22=110= 5 \times 22 = 110 units. Remaining work =120−110=10= 120 - 110 = 10 units.

  • Day 16 (Day 1 of 6th cycle, A + B): They do 9 units. Remaining work =1= 1 unit.
  • Day 17 (Day 2 of 6th cycle, B + C): They complete the remaining 1 unit.

Alex works on Days 1 and 3 of each cycle, and on Day 16 (Day 1 of the 6th cycle). In 5 complete cycles, Alex works 5×2=105 \times 2 = 10 days. Plus Day 16, Alex works 1 more day. Total days Alex works =10+1=11= 10 + 1 = 11 days.

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