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S99-16

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S99-16 [#permalink]

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New post 16 Sep 2014, 01:53
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Question Stats:

52% (02:40) correct 48% (02:34) wrong based on 65 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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Re S99-16 [#permalink]

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New post 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.


Answer: C
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Re: S99-16 [#permalink]

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New post 22 Oct 2016, 20:48
any other quick solution to this.......
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Re: S99-16 [#permalink]

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New post 23 Oct 2016, 00:39
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 \(d=10\).

Answer: C.
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Re: S99-16 [#permalink]

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New post 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?
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Re: S99-16 [#permalink]

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New post 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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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: S99-16   [#permalink] 24 Jan 2018, 08:07
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