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MariettaO [177]
3 years ago
6

Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating simultaneously, can fill a certain tank i

n 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
Mathematics
1 answer:
jeka943 years ago
8 0

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

\begin{cases}\dfrac1a+\dfrac1b=\dfrac1{\frac65}=\dfrac56\\\\\dfrac1a+\dfrac1c=\dfrac1{\frac32}=\dfrac23\\\\\dfrac1b+\dfrac1c=\dfrac12\end{cases}

Now,

\left(\dfrac1a+\dfrac1b\right)-\left(\dfrac1a+\dfrac1c\right)=\dfrac56-\dfrac23\implies\dfrac1b-\dfrac1c=\dfrac16

Then

\left(\dfrac1b+\dfrac1c\right)+\left(\dfrac1b-\dfrac1c\right)=\dfrac12+\dfrac16\implies\dfrac2b=\dfrac23\implies b=3

This means

\dfrac1a+\dfrac13=\dfrac56\implies\dfrac1a=\dfrac12\implies a=2

and

\dfrac13+\dfrac1c=\dfrac12\implies\dfrac1c=\dfrac16\implies c=6

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

\dfrac1a+\dfrac1b+\dfrac1c=1

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

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