Question

Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating simultaneously, can fill a certain tank in 6/5 hours; pumps A and C, operating simultaneously, can fill the tank in 3/2 hours; and pumps B and C, operating simultaneously, can fill the tank in 2 hours. How many hours does it take pumps A, B, and C, operating simultaneously, to fill the tank.A. 1/3B. 1/2C. 1/4 D. 1E. 5/6

Answer 1

Let
`a,b,c` be the time it takes for pumps A, B, and C (respectively) to fill 1 tank. Then

`\begin{cases}\frac1a+\frac1b=\frac1{\frac65}=\frac56\\\\\frac1a+\frac1c=\frac1{\frac32}=\frac23\\\\\frac1b+\frac1c=\frac12\end{cases}`

Now,

`\left(\frac1a+\frac1b\right)-\left(\frac1a+\frac1c\right)=\frac56-\frac23\implies\frac1b-\frac1c=\frac16`

Then

`\left(\frac1b+\frac1c\right)+\left(\frac1b-\frac1c\right)=\frac12+\frac16\implies\frac2b=\frac23\implies b=3`

This means

`\frac1a+\frac13=\frac56\implies\frac1a=\frac12\implies a=2`

and

`\frac13+\frac1c=\frac12\implies\frac1c=\frac16\implies c=6`

Working together, all 3 pumps would operate at a rate of

`\frac1a+\frac1b+\frac1c=1`

or 1 tank in 1 hour, so the answer is D.