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# S99-16

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Math Expert
Joined: 02 Sep 2009
Posts: 49915

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16 Sep 2014, 01:53
00:00

Difficulty:

85% (hard)

Question Stats:

52% (02:40) correct 48% (02:38) wrong based on 67 sessions

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A bathtub has two faucets, P and Q, and one drain. Faucet P alone can fill the whole tub in ten minutes, and faucet Q alone can fill the whole tub four minutes faster than the drain can empty the whole tub. With faucets P and Q both running and the drain unstopped, the tub fills in six minutes. How long would the drain take to empty the whole tub?

A. 5 and $$\frac{5}{11}$$ minutes
B. 6 minutes
C. 10 minutes
D. 19 minutes
E. 30 minutes

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Joined: 02 Sep 2009
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16 Sep 2014, 01:53
Official Solution:

A bathtub has two faucets, P and Q, and one drain. Faucet P alone can fill the whole tub in ten minutes, and faucet Q alone can fill the whole tub four minutes faster than the drain can empty the whole tub. With faucets P and Q both running and the drain unstopped, the tub fills in six minutes. How long would the drain take to empty the whole tub?

A. 5 and $$\frac{5}{11}$$ minutes
B. 6 minutes
C. 10 minutes
D. 19 minutes
E. 30 minutes

Let $$t$$ stand for the desired time, so that the drain can empty the tub in $$t$$ minutes and faucet Q can fill the tub in $$(t - 4)$$ minutes. Also, the drain does negative work as it empties the tub. The rates for the pipes and the drain are thus

P: 1 tub in $$10$$ min $$= \frac{1}{10}$$ tub per min

Q: 1 tub in $$(t - 4)$$ min $$= \frac{1}{t - 4}$$ tub per min

Drain: -1 tub in $$t$$ min $$= -\frac{1}{t}$$ tub per min

Using the fact that all three fixtures together take 6 minutes to fill 1 tub, set up an RTW chart, and use the chart to calculate the total work quantities in the last column.
Rate (tub/min) $$\times$$ Time (min) $$=$$ Total Work (tubs) P $$\frac{1}{10}$$ $$\times$$ 6 $$=$$ $$\frac{3}{5}$$ Q $$\frac{1}{t - 4}$$ $$\times$$ 6 $$=$$ $$\frac{6}{t - 4}$$ Drain $$\frac{-1}{t}$$ $$\times$$ 6 $$=$$ $$\frac{-6}{t}$$ Total n/a n/a 1
Set up an equation summing up the work:
$$\frac{3}{5} + \frac{6}{t - 4} - \frac{6}{t} = 1$$

Multiply by the common denominator, $$5t(t - 4)$$:
$$3t(t - 4) + 6(5t) - 6(5)(t - 4) = 5t(t - 4)$$
$$3t^2 - 12t + 30t - 30t + 120 = 5t^2 - 20t$$
$$0 = 2t^2 - 8t - 120$$
$$0 = t^2 - 4t - 60$$
$$0 = (t - 10)(t + 6)$$

$$t = 10$$ or $$t = -6$$

The negative value is absurd, so $$t = 10$$; the drain can empty the tub in 10 minutes.

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22 Oct 2016, 20:48
any other quick solution to this.......
Math Expert
Joined: 02 Sep 2009
Posts: 49915

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23 Oct 2016, 00:39
1
VIJAYTHAPLIYAL wrote:
any other quick solution to this.......

A bathtub has two faucets, P and Q, and one drain. Faucet P alone can fill the whole tub in ten minutes, and faucet Q alone can fill the whole tub four minutes faster than the drain can empty the whole tub. With faucets P and Q both running and the drain unstopped, the tub fills in six minutes. How long would the drain take to empty the whole tub?

A. 5 and $$\frac{5}{11}$$ minutes
B. 6 minutes
C. 10 minutes
D. 19 minutes
E. 30 minutes

Let $$p$$ and $$q$$ be the times in minutes needed for faucets P and Q working alone to fill the tub and d be the time in minutes needed for drain to empty the tub.

Given:
Faucet P alone can fill the whole tub in ten minutes --> $$p=10$$;
Faucet Q alone can fill the whole tub four minutes faster than the drain can empty the whole tub --> $$q=d-4$$;
Faucets P and Q both running and the drain unstopped, the tub fills in six minutes --> $$\frac{1}{p}+\frac{1}{q}-\frac{1}{d}=\frac{1}{6}$$ --> $$\frac{1}{10}+\frac{1}{d-4}-\frac{1}{d}=\frac{1}{6}$$ --> $$\frac{1}{d-4}-\frac{1}{d}=\frac{1}{15}$$ --> substituting the values from the answer choices we'll get $$d=10$$.

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24 Jan 2018, 07:51
The question is not clear. The question should be "How long would the drain take to empty the whole tub WHEN BOTH FAUCETS ARE ON". Am I wrong?
Math Expert
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24 Jan 2018, 08:07
susig27 wrote:
The question is not clear. The question should be "How long would the drain take to empty the whole tub WHEN BOTH FAUCETS ARE ON". Am I wrong?

The question is: How long would the drain take to empty the whole tub, when none of the faucets is working?

Check solution here: https://gmatclub.com/forum/s99-184744.html#p1752472
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14 Oct 2018, 03:48
susig27 wrote:
The question is not clear. The question should be "How long would the drain take to empty the whole tub WHEN BOTH FAUCETS ARE ON". Am I wrong?

You probably are.
maybe you did not thoroughly read the second part of the passage! I am claiming this just 'cause did the same mistake at the beginning. I did the equation considering a "stopped" drain ( so just the combination of the faucets' rates, without considering the drain).
In realty the text says "unstopped" so the equation have to be
1/x + 1/10 - 1/(4+x) = 1/6
solving it you get x=6 (i considered 1/x the rate of P and 1/(x+4) the drain's rate)
Hope it helps!
Re: S99-16 &nbs [#permalink] 14 Oct 2018, 03:48
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# S99-16

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