Question : A and B working together can do a piece of work in $4\frac{1}{2}$ hours. B and C working together can do it in 3 hours. C and A working together can do it in $2\frac{1}{4}$ hours. All of them begin the work at the same time. Find how much time they will take to finish the piece of work.
Option 1: 3 hours
Option 2: 2 hours
Option 3: 2.5 hours
Option 4: 3.25 hours
Correct Answer: 2 hours
Solution : Amount of work done by A and B in 1 hour = $\frac{2}{9}$ Amount of work done by B and C in 1 hour = $\frac{1}{3}$ Amount of work done by A and C in 1 hour = $\frac{4}{9}$ Adding all the equations, we have: ⇒ 2 × (A + B + C)'s 1 hour work = $\frac{2}{9}+\frac{1}{3}+\frac{4}{9}$ ⇒ 2 × (A + B + C)'s 1 hour work = $\frac{2+3+4}{9}=1$ ⇒ (A + B + C)'s 1 hour work = $\frac{1}{2}$ (A + B + C) can finish $\frac{1}{2}$ work in 1 hour. (A + B + C) can finish the total work in 2 hours. Hence, the correct answer is 2 hours.
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