Time and Work Question 2 - CAT 2024
Sam can complete a job in 20 days when working alone. Mohit is twice as fast as Sam and thrice as fast as Ayna in the same job. They undertake a job with an arrangement where Sam and Mohit work together on the first day, Sam and Ayna on the second day, Mohit and Ayna on the third day, and this three-day pattern is repeated till the work gets completed. Then, the fraction of total work done by Sam is:
Step-by-Step Explanation
- Determine the Time Needed for Each Worker:
Sam alone = 20 days.
Mohit is twice as fast as Sam ⟹ 10 days.
Mohit is thrice as fast as Ayna ⟹ Ayna takes 10 × 3 = 30 days.
- Assign a Smart Value for Total Work:
Let the total work be the LCM of 20, 10, and 30, which is 60 units.
Sam's efficiency = 60/20 = 3 units/day
Mohit's efficiency = 60/10 = 6 units/day
Ayna's efficiency = 60/30 = 2 units/day
- Analyze the 3-Day Cycle:
Day 1 (Sam + Mohit): 3 + 6 = 9 units
Day 2 (Sam + Ayna): 3 + 2 = 5 units
Day 3 (Mohit + Ayna): 6 + 2 = 8 units
Work done in one 3-day cycle = 9 + 5 + 8 = 22 units
- Track the Cycles to Completion:
After 2 complete cycles (6 days total):
22 × 2 = 44 units completed.
Remaining work = 60 − 44 = 16 units.
Day 7 (Sam + Mohit): They complete 9 units.
Remaining work = 16 − 9 = 7 units.
Day 8 (Sam + Ayna): They complete 5 units.
Remaining work = 7 − 5 = 2 units.
Day 9 (Mohit + Ayna): The remaining 2 units are finished.
- Calculate Sam's Contribution:
Sam works on Day 1 and Day 2 of each cycle, as well as on Day 7 and Day 8.
Total days Sam worked = 2 + 2 + 1 + 1 = 6 days
Total work done by Sam = 6 days × 3 units/day = 18 units
Fraction of work done by Sam = 18/60 = 3/10
Correct Option: B
💡 FundaMakers Tip:
For periodic rotation schedules, calculate the total output per single complete block rather than writing out a day-by-day table. Once you determine the output of the cycle block (22 units), use simple division to locate the boundary wall, and manually check only the final remaining units.