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Time and Work with Alternating Shifts

CAT 2024 Slot 2 · Quantitative Ability · Hard · Arithmetic

This is a hard Quantitative Ability question from the CAT 2024 Slot 2 paper. It tests Arithmetic. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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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

Answer

139

Explanation

Let total work = 300 units. Let rates of Amal, Vimal, and Sunil be A,V,SA, V, S units/day respectively. A+V=300150=2A + V = \frac{300}{150} = 2 V+S=300100=3V + S = \frac{300}{100} = 3

From the given work breakdown: 75A+135V+45S=30075A + 135V + 45S = 300 75(A+V)+60V+45S=30075(A + V) + 60V + 45S = 300 75(2)+15V+45(V+S)=30075(2) + 15V + 45(V + S) = 300 150+15V+45(3)=300150 + 15V + 45(3) = 300 150+15V+135=300  ⟹  15V=15  ⟹  V=1 unit/day150 + 15V + 135 = 300 \implies 15V = 15 \implies V = 1\text{ unit/day}

Hence: A=2−1=1 unit/dayA = 2 - 1 = 1\text{ unit/day} S=3−1=2 units/dayS = 3 - 1 = 2\text{ units/day}

In a 6-day cycle under the new schedule:

  • Amal works all 6 days: 6×1=66 \times 1 = 6 units.
  • Vimal works on days 2, 4, 6 (3 days): 3×1=33 \times 1 = 3 units.
  • Sunil works on days 3, 6 (2 days): 2×2=42 \times 2 = 4 units. Total work done in 6 days = 6+3+4=136 + 3 + 4 = 13 units.

Number of 6-day cycles to get close to 300 units: ⌊300/13⌋=23 cycles\lfloor 300 / 13 \rfloor = 23\text{ cycles} Work in 23 cycles (23×6=13823 \times 6 = 138 days) =23×13=299= 23 \times 13 = 299 units. Remaining work =300−299=1= 300 - 299 = 1 unit.

On day 139 (which is day 1 of cycle 24): Amal works alone and does 1 unit of work, completing the task. Total days required = 139.

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