Time and Work Question 3 - CAT 2024
Amal and Vimal together can complete a task in 150 days, while Vimal and Sunil together can complete the same task in 100 days. Amal starts working on the task and works for 75 days, then Vimal takes over and works for 135 days. Finally, Sunil takes over and completes the remaining task in 45 days. If Amal had started the task alone and worked on all days, Vimal had worked on every second day, and Sunil had worked on every third day, then the number of days required to complete the task would have been:
Step-by-Step Explanation
Let the individual daily work efficiencies of Amal, Vimal, and Sunil be A, V, and S, respectively.
- Set Up the Baseline Equations:
Let the total work be W.
150A + 150V = W
⟹ A + V = W/150 --- (Equation 1)
100V + 100S = W
⟹ V + S = W/100 --- (Equation 2)
- Analyze the Alternating Shift Scenario:
Amal works for 75 days, Vimal works for 135 days, and Sunil works for 45 days:
75A + 135V + 45S = W
Rearrange this expression to use our combined pair rates:
75(A + V) + 60V + 45S = W
Substitute Equation 1 into the expression:
75(W/150) + 60V + 45S = W
⟹ W/2 + 60V + 45S = W
60V + 45S = W/2
⟹ 120V + 90S = W --- (Equation 3)
- Solve for the Efficiencies:
Equate Equation 2 (scaled by 3) and Equation 3:
300V + 300S = 3W
and
360V + 270S = 3W
300V + 300S = 360V + 270S
⟹ 30S = 60V
⟹ S = 2V
Substitute S = 2V into Equation 2:
100V + 100(2V) = W
⟹ 300V = W
Let us assume a convenient total workload value of W = 300 units.
V = 1 unit/day
S = 2 units/day
From A + V = 300/150 = 2:
A = 1 unit/day
- Analyze the Final Work Cycle:
Amal works every day, Vimal works every 2nd day, and Sunil works every 3rd day.
The LCM of their work periods is 6 days, so we evaluate a 6-day cycle:
- Amal works all 6 days: 6 × 1 = 6 units
- Vimal works 3 days: 3 × 1 = 3 units
- Sunil works 2 days: 2 × 2 = 4 units
Total work completed in a 6-day cycle = 6 + 3 + 4 = 13 units
To reach the total target workload of 300 units:
Number of complete cycles = ⌊300/13⌋ = 23 cycles
Work done in 23 cycles = 23 × 13 = 299 units
Days elapsed = 23 × 6 = 138 days
Remaining Work = 300 − 299 = 1 unit
On the 139th day (Day 1 of a new cycle), only Amal is scheduled to work. Since his efficiency is exactly 1 unit/day, he completes the remaining 1 unit of work in 1 day.
Total Time = 138 + 1 = 139 days
Correct Answer: 139
💡 FundaMakers Tip:
When handling partial overlapping schedules like "A for 75 days, then B for 135 days," group the days into blocks that match the given joint efficiency rates (e.g., pairing 75 days of A with 75 days of B). This lets you use the fractional pairs data directly, turning a complicated system of equations into a straightforward comparison.