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# M26-23

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Math Expert
Joined: 02 Sep 2009
Posts: 47946

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16 Sep 2014, 01:25
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Difficulty:

55% (hard)

Question Stats:

76% (01:50) correct 24% (01:32) wrong based on 37 sessions

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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. $$\frac{yz+xz-xy}{yz}$$

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Joined: 02 Sep 2009
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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.

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Joined: 12 Oct 2015
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GMAT 1: 740 Q50 V40

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24 Mar 2016, 11:57
1
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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Joined: 02 Sep 2009
Posts: 47946

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25 Mar 2016, 04:16
thomasgaar18 wrote:
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?

Correct. Edited. Thank you for noticing.
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Joined: 18 May 2016
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15 Dec 2017, 22:23
1
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

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
Kudos?
M26-23 &nbs [#permalink] 15 Dec 2017, 22:23
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# M26-23

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