M26-23

Question

A swimming pool has two water pumps, A and B, and one drain, C. Pump A can fill the empty pool by itself in \(x\) hours, while pump B can do so in \(y\) hours. The drain C can empty the entire pool in \(z\) hours, with \(z > x\). When both pumps A and B are operating simultaneously and drain C is open until the pool is filled, which of the following represents the fraction of the total pool volume added by pump A until the pool is filled?

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. \(\frac{yz+xz-xy}{yz}\)

Show Answer

Official Answer

B

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}

Bunuel · Accepted Answer

Official Solution:

A swimming pool has two water pumps, A and B, and one drain, C. Pump A can fill the empty pool by itself in \(x\) hours, while pump B can do so in \(y\) hours. The drain C can empty the entire pool in \(z\) hours, with \(z > x\). When both pumps A and B are operating simultaneously and drain C is open until the pool is filled, which of the following represents the fraction of the total pool volume added by pump A until the pool is filled?

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. \(\frac{yz+xz-xy}{yz}\)

Show Answer

Official Answer

B

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}

When both pumps A and B are running, and the drain is open, the pool fills at a rate of  \frac{1}{x} + \frac{1}{y} - \frac{1}{z} = \frac{yz + xz - xy}{xyz}  pools per hour. Therefore, it takes  \frac{xyz}{yz + xz - xy}  hours to fill the pool (since time is the reciprocal of rate).
 
In  \frac{xyz}{yz+xz-xy}  hours, pump A will contribute  rate * time = \frac{1}{x} * \frac{xyz}{yz+xz-xy} = \frac{yz}{yz+xz-xy}  of the total pool volume.
 
For instance, suppose the pool will be filled in 10 hours, and pump A alone can fill the entire pool in 20 hours. In 10 hours, pump A will pump  \frac{1}{20} * 10 = \frac{1}{2}  of the total pool volume.

Answer: B

ppnimkar · Answer

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}

RenanBragion · Answer

I think this is a  high-quality  question.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Jessie265 · Answer

A can fill the empty pool by itself in x hours => A can fill V/x per hour
B can fill the empty pool by itself in y hours => B can fill V/y per hour
C can empty the entire pool by itself in z hours => 1 hour can drain V/z per hour 
A, B, C do work simultaneously, the pool will be fill in t => V=(V/x+V/y-V/z)*t
The amount of water A can fill in t is V/x*t
=> The fraction will be:
((V/x)*t)/((V/x+V/y-V/z)*t)=>B

PReciSioN · Answer

Hi  Bunuel ,

The official answer correctly reflects the  amount of water pumped by A in filling the pool . However, the question asks us to find t he fraction of water that was pumped into the pool by pump A.

This fraction = water pumped into the pool by A/ (water pumped into the pool by A + water pumped into the pool by B)

*please note that the drain does not contribute to pumping in of water and so is not considered in this fraction.

This would then equal to   /   = yz/ (yz+xz) = y/(x+y).

Also please note that if x=y=z that is all have the same rate. then naturally A should pump in half of the water and B should pump in half of the water. The official answer in such a case would suggest that the fraction is 1 which is not the case.

Bunuel · Answer

Rephrased the last sentence of the question to address this ambiguity. Thank you!

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

M26-23

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards