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Math Expert V
Joined: 02 Sep 2009
Posts: 58418

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Difficulty:   85% (hard)

Question Stats: 54% (02:47) correct 46% (03:04) wrong based on 63 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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Math Expert V
Joined: 02 Sep 2009
Posts: 58418

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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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Intern  Joined: 02 Oct 2014
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any other quick solution to this.......
Math Expert V
Joined: 02 Sep 2009
Posts: 58418

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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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Intern  Joined: 18 Jan 2018
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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 V
Joined: 02 Sep 2009
Posts: 58418

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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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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!
Intern  B
Joined: 10 Feb 2017
Posts: 47
Location: India
Schools: Rotman '20
GMAT 1: 710 Q49 V37 GPA: 4

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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?

go with the most easy choice that you can handle

always choose LCM method

P takes 10 mins
From the option you take for ex 6(option-b) for drain D
SO Q MUST TAKE 6-4=2 MINS
so now find LCM(10,6,2)=30
FOR MIN P WILL DO 30/10=3(SINCE WORK IS NOT 1 NOW,ITS 30 NOW)
Q=15 WORKS PER MIN
D=5 WORKS/MIN

ALL TOGETHER TOTAL WORK WILL BE =3+15-5=13/MIN.

NOW LOOK 13 WORKS*6 MINS (WHY 6 IS MULTIPLIED,BKZ ITS GIVEN IN THE QN THAT WHEN ALL WORK TOGETHER THEIR WAY IT TAKES 6 MINS TO FEEL THE TANK)
BUT ITS NOT,SO OPTION -B OUT
NEXT OPTION C
P=10
D=10(OPTION)
Q=10-4=6
LCM(10,10,6)=30
PER MIN WORK=3-3+5=5
NOW OUT OF 30 WORKS

IF ALL WORKING THEIR WAY AND 5 IS THE NET WORK PER MINUTE,CAN THEY FEEL THE TANK IN 6 MINUTES?
ANSWER IS YES 5*6=30(TOTAL Capacity as per our lcm work out.

think rethink practice and get the correct choice as dont forget to help for 1 kudo.
Intern  B
Joined: 03 Nov 2018
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Why can't I get the solution with the combined work formula? T = A*B / A +B
This would be:
(10*(t-4) / (10 + t-4)) - t = 6
Intern  B
Joined: 14 Jan 2019
Posts: 3

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The easiest solution is to set up the rate equation and plug in the answer choices;

1/10 + 1/t-4 - 1/t = 1/6

Plugging t = 10 gives you 1/6 and that is your correct answer. Re: S99-16   [#permalink] 30 Jul 2019, 11:51
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