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A can do a piece of work in 80 days; B and C together can do it in 40 days, while A and C together can do it in 32 days. How long (in days) will B alone take to do it?
120
140
160
180
160
Calculate the LCM of 80, 40, and 32, which is 160. Calculate individual efficiencies: E(A)=2, E(B+C)=4, E(A+C)=5, then subtract to find E(B).
A takes 80 days, B and C take 40 days, and A and C take 32 days to complete the work.
Efficiency=TimeTotal┬аWorkтАЛ
Calculate the LCM of 80, 40, and 32, which is 160. Calculate individual efficiencies: E(A)=2, E(B+C)=4, E(A+C)=5, then subtract to find E(B).
Students often add (A+C) and (B+C) directly without realizing that A's efficiency must be subtracted from the combined efficiency of (A+C) to isolate C's work before finding B.
Calculate Total Work
Assume Total Work as the Least Common Multiple (LCM) of 80, 40, and 32, which is 160 units.
LCM(80,40,32)=160
Calculate Individual Efficiencies
Efficiency is defined as work divided by time. For A: 160/80=2. For B+C: 160/40=4. For A+C: 160/32=5.
EAтАЛ=2,EB+CтАЛ=4,EA+CтАЛ=5
Isolate B's Efficiency
First find C's efficiency from A+C: ECтАЛ=EA+CтАЛтИТEAтАЛ=5тИТ2=3. Then find B's efficiency from B+C: EBтАЛ=EB+CтАЛтИТECтАЛ=4тИТ3=1.
EBтАЛ=4тИТ(5тИТ2)=1
Calculate Time for B
Time taken by B alone is Total Work divided by B's efficiency.
TimeBтАЛ=1160тАЛ=160
C is correct because B's efficiency is 1 unit/day, requiring 160 days to complete 160 units of work.
This method of finding individual components from group efficiencies is identical to solving systems of linear equations in Algebra.
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