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!

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M26-23

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