Time and Work Question 3 - CAT 2022
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:
Step-by-Step Explanation
- Establish Relative Daily Efficiencies:
Alex is twice as fast as Bob, and thrice as fast as Cole.
Let us assign Alex an integer daily efficiency value based on the common multiple of 2 and 3, which is 6 units/day.
Bob's efficiency = 6/2 = 3 units/day
Cole's efficiency = 6/3 = 2 units/day
- Determine Total Workload:
Bob finishes the job alone in 40 days:
Total Work = 40 days × 3 units/day
= 120 units
- Analyze the 3-Day Roster Cycle:
Day 1 (Alex + Bob): 6 + 3 = 9 units
Day 2 (Bob + Cole): 3 + 2 = 5 units
Day 3 (Cole + Alex): 2 + 6 = 8 units
Work completed in one 3-day block = 9 + 5 + 8 = 22 units
- Track Cycles to Completion:
Number of complete 3-day blocks = ⌊120/22⌋ = 5 blocks.
Work done in 5 blocks (15 days) = 5 × 22 = 110 units
Remaining Work = 120 − 110 = 10 units
Day 16 (Alex + Bob roster): They complete 9 units.
Remaining work = 10 − 9 = 1 unit.
Day 17 (Bob + Cole roster): They easily complete the final 1 unit, finishing the job.
- Calculate Alex's Total Active Days:
Alex works exactly 2 days out of every 3-day block (Day 1 and Day 3).
Days worked during the first 5 blocks = 5 × 2 = 10 days.
Alex is also active on Day 16 (Alex + Bob roster step) = 1 day.
Alex is not active on Day 17 (Bob + Cole roster step).
Total Days Alex worked = 10 + 1 = 11 days
Correct Answer: 11
💡 FundaMakers Tip:
When tracking rotational shift setups, make sure to isolate the individual presence of the target person during the final leftover days. A common trap is calculating the total time the project takes (17 days) and scaling it down proportionally, which leads to off-by-one errors. Check the final days step-by-step.