Question

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 ha

Accepted Answer

139. Let total work = 300 units. Let rates of Amal, Vimal, and Sunil be $A, V, S$ units/day respectively. $$A + V = \frac{300}{150} = 2$$ $$V + S = \frac{300}{100} = 3$$ From the given work breakdown: $$75A + 135V + 45S = 300$$ $$75(A + V) + 60V + 45S = 300$$ $$75(2) + 15V + 45(V + S) = 300$$ $$150 + 15V + 45(3) = 300$$ $$150 + 15V + 135 = 300 \implies 15V = 15 \implies V = 1	ext{ unit/day}$$ Hence: $$A = 2 - 1 = 1	ext{ unit/day}$$ $$S = 3 - 1 = 2	ext{ units/day}$$ In a 6-day cycle under the new schedule: - Amal works all 6 days: $6 	imes 1 = 6$ units. - Vimal works on days 2, 4, 6 (3 days): $3 	imes 1 = 3$ units. - Sunil works on days 3, 6 (2 days): $2 	imes 2 = 4$ units. Total work done in 6 days = $6 + 3 + 4 = 13$ units. Number of 6-day cycles to get close to 300 units: $$\lfloor 300 / 13 floor = 23	ext{ cycles}$$ Work in 23 cycles ($23 	imes 6 = 138$ days) $= 23 	imes 13 = 299$

Time and Work with Alternating Shifts

Explanation

Practise this under exam conditions