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A can do a certain work in the same time in which B and C together can do it. If A and B together could do it in 12 days and C alone in 24 days, then B alone could do it in:
24 days
48 days
36 days
72 days
48 days
Use the LCM of 12 and 24 which is 24 units. Determine individual efficiencies, solve the linear system by equating total work rate of A with B+C.
Time for A+B = 12 days, time for C = 24 days. A completes work in same time as B+C.
Efficiency=TimeTotal┬аWorkтАЛ
Use the LCM of 12 and 24 which is 24 units. Determine individual efficiencies, solve the linear system by equating total work rate of A with B+C.
Students often assume A, B, and C have equal efficiencies or confuse the A = B+C relation with A+B+C = 1.
Define efficiencies
Let total work be the LCM of 12 and 24, which is 24 units. Thus, efficiency of A+B is 24/12=2 units/day and efficiency of C is 24/24=1 unit/day.
EAтАЛ+EBтАЛ=2,ECтАЛ=1
Use the A = B+C condition
We are given that A takes the same time as B+C, meaning their efficiencies are equal: EAтАЛ=EBтАЛ+ECтАЛ.
EAтАЛтИТEBтАЛ=ECтАЛ=1
Solve for EBтАЛ
We have a system of two equations: EAтАЛ+EBтАЛ=2 and EAтАЛтИТEBтАЛ=1. Subtracting the second from the first gives 2EBтАЛ=1, so EBтАЛ=0.5 units/day.
2EBтАЛ=1тЯ╣EBтАЛ=0.5
Calculate time for B
Time taken by B = Total work / Efficiency of B, which is 24/0.5=48.
TimeBтАЛ=0.524тАЛ=48
B is correct because the calculation of B's efficiency (0.5 units/day) leads to a total time of 48 days.
This logic is identical to mixture and allegation problems where the sum of two rates equals a third rate; always look for the sum of variables.
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