Question: 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 > x. with pumps A and B both running and the drain C unstopped till the pool is filled, which fo the following represents the amount of water in terms of the fraction of the pool which pump A pumped into the pool?​

1.  \(\frac{xy}{x+y+z}\)
2. \(\frac{yz}{yz+xz-xy}\)
3. \(\frac{yz}{yz+xz+xy}\)
4. \(\frac{xyz}{yz+xz-xy}\)
5. \(\frac{yz+xz-xy}{yz}\)

Answer:
Solution with Explanation:
Approach Solution (1):

With pumps A and B both running and the drain unstopped the pool will be filled in a rate \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}= \frac{yz+zx-xy}{xyz}\)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 intotal he pool

Correct Option: B

Approach Solution (2):

Let x = 2, y = 3 and z = 6 as the number of hours it takes A and B to fill and C to drain the pool respectively. (z > x)

Hence the rates for A = \(\frac{1}{2}\), B =\(\frac{1}{3}\), C =\(\frac{1}{6}\)

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

\(\frac{1}{2} /\frac{1}{3}\)

\(\frac{3}{4}\)filled by A

Substitute in answers:

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

Correct option: B

“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 > x. with pumps A and B both running and the drain C unstopped till the pool is filled, which fo the following represents the amount of water in terms of the fraction of the pool which pump A pumped into the pool?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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