# Calculate work of machines

This is a GRE question:

Working together, machines A, B, and C can complete a job in 24 hours. Working together, machines B, C, and D can complete the job in 60 hours. If working alone, machine D can complete the job in 120 hours, in how many hours can machine A complete the job when working alone?

Options for the answer are: $16\frac{2}{3}$, $24$, $28$, $30$, and $33\frac{1}{3}$. The correct answer is $30$. Why?

Hint: Let $A$, $B$, $C$, $D$ represent the number of jobs each machine can do individually per 120 hours.
Then: $$A+B+C={120 \over 24}=5$$ $$B+C+D={120 \over 60}=2$$ $$D={120 \over 120}=1$$
Add the first and third equations to yield $A+B+C+D=6$. Then subtract the second equation to yield $A=4$. So machine A does 4 jobs in 120 hours, so one job takes 30 hours.
• Thanks, @Fengyang Wang. How did you come up with the three equations above? In other words, why is $A + B + C = 5$, and $B + C + D = 2$, and $D = 1$? It's been over 15 years since my last math class, so I'm extra rusty. – Alex Apr 26 '14 at 3:57
• @Alex, note that if A, B, and C are working together, then their combined work for 120 hours is $A+B+C$, since each letter represents an individual's work over that span. Since we know they do one job in 24 hours, it remains to divide 120 by 24 to see how many jobs can be done in 120 hours by them combined. The result is $A+B+C=5$. – Fengyang Wang Apr 26 '14 at 4:00