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M26-23

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M26-23 [#permalink]

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New post 16 Sep 2014, 01:25
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A pool has two water pumps A and B and one drain C. Pump A alone can fill the whole pool in \(x\) hours, and pump B alone can fill the whole pool in \(y\) hours. The drain can empty the whole pool in \(z\) hours, where \(z \gt x\). With pumps A and B both running and the drain C unstopped till the pool is filled, which of the following represents the amount of water in terms of the fraction of the pool which pump A pumped into the pool?

A. \(\frac{yz}{x+y+z}\)
B. \(\frac{yz}{yz+xz-xy}\)
C. \(\frac{yz}{yz+xz+xy}\)
D. \(\frac{xyz}{yz+xz-xy}\)
E. \(\frac{yz+xz-xy}{yz}\)
[Reveal] Spoiler: OA

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Re M26-23 [#permalink]

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New post 16 Sep 2014, 01:25
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Official Solution:

A pool has two water pumps A and B and one drain C. Pump A alone can fill the whole pool in \(x\) hours, and pump B alone can fill the whole pool in \(y\) hours. The drain can empty the whole pool in \(z\) hours, where \(z \gt x\). With pumps A and B both running and the drain C unstopped till the pool is filled, which of the following represents the amount of water in terms of the fraction of the pool which pump A pumped into the pool?

A. \(\frac{yz}{x+y+z}\)
B. \(\frac{yz}{yz+xz-xy}\)
C. \(\frac{yz}{yz+xz+xy}\)
D. \(\frac{xyz}{yz+xz-xy}\)
E. \(\frac{yz+xz-xy}{yz}\)


With pumps A and B both running and the drain unstopped the pool will be filled in a rate \(\frac{1}{x}+\frac{1}{y}-\frac{1}{z}=\frac{yz+xz-xy}{xyz}\) pool/hour. So, the pool will be filled in \(\frac{xyz}{yz+xz-xy}\) hours (time is reciprocal of rate).

In \(\frac{xyz}{yz+xz-xy}\) hours A will pump \(\frac{1}{x}*\frac{xyz}{yz+xz-xy}=\frac{yz}{yz+xz-xy}\) amount of the water into the pool.


Answer: B
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Re: M26-23 [#permalink]

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New post 24 Mar 2016, 11:57
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Hello!

I think in the first fraction in the solution it should be yz+xz−xy and not yz+xz−zy in the nominator or am i wrong?
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Re: M26-23 [#permalink]

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New post 25 Mar 2016, 04:16
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M26-23 [#permalink]

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New post 15 Dec 2017, 22:23
Hello Bunuel

I used a slightly different approach

Noting z > x
Let x = 2, y = 3 & z = 6 as the number of hours it takes A & B to fill and C to drain the pool respectively
Hence the rates for A = 1/2, B = 1/3 and C = 1/6

Together they fill
\(\frac{1}{x} + \frac{1}{y} - \frac{1}{z}\)
\(\frac{1}{2} + \frac{1}{3} - \frac{1}{6}\)
\(\frac{3+2-1}{6}\)
\(\frac{2}{3}\) rate/hour

Fraction of A/entire rate
\(\frac{1}{2}\)/\(\frac{2}{3}\)
\(\frac{3}{4}\) filled by A

Substitute in answers:
B) \(\frac{yz}{(yz+xz-xy)} = \frac{3*6}{(3*6)+(2*6)-(2*3)} = \frac{18}{18+12-6} = \frac{3}{4}\)

Please correct me if I'm wrong
Thanks
Prathamesh
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M26-23   [#permalink] 15 Dec 2017, 22:23
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