Two taps can fill a cistern in 20 minutes and 30 minutes...

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BTGmoderatorLU: Moderator; Posts: 2505; Joined: Sun Oct 15, 2017 1:50 pm; Followed by:6 members

Two taps can fill a cistern in 20 minutes and 30 minutes...

by BTGmoderatorLU » Thu Feb 08, 2018 7:06 am

Two taps can fill a cistern in 20 minutes and 30 minutes. The first tap was opened initially for x minutes after which the second tap was opened. If it took a total of 15 minutes for the tank to be filled, what is the value of x?

A. 5.0
B. 7.5
C. 9.0
D. 10.0
E. 12.5

The OA is B.

I know the rate of the first tap is 1/20 minutes, and the rate of the second tap is 1/30.

Then, the first was working x/20 minutes, and after which the second tap was opened.

The total time was 15 minutes. I stuck here.

I'm confused by this PS question. Experts, any suggestion? Thanks in advance.

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Source: Beat The GMAT — Problem Solving | Thu Feb 08, 2018 7:06 am

EconomistGMATTutor: GMAT Instructor; Posts: 555; Joined: Wed Oct 04, 2017 4:18 pm; Thanked: 180 times; Followed by:12 members

answer

by EconomistGMATTutor » Thu Feb 08, 2018 8:00 am

LUANDATO wrote:Two taps can fill a cistern in 20 minutes and 30 minutes. The first tap was opened initially for x minutes after which the second tap was opened. If it took a total of 15 minutes for the tank to be filled, what is the value of x?

A. 5.0
B. 7.5
C. 9.0
D. 10.0
E. 12.5

The OA is B.

I know the rate of the first tap is 1/20 minutes, and the rate of the second tap is 1/30.

Then, the first was working x/20 minutes, and after which the second tap was opened.

The total time was 15 minutes. I stuck here.

I'm confused by this PS question. Experts, any suggestion? Thanks in advance.

Hi LUANDATO.

Let's continue your explanation.

The first tap rate is 1/20 per minute.

The second tap rate is 1/30 per minute.

Now, we have to set the following equation: $$rate\ tap\ 1\ during\ x\ \text{minutes}\ +\ \left(time\ both\ taps\ worked\ together\right)\left(rate\ tap\ 1\ +\ rate\ tap\ 2\right)$$ $$=\ whole\ \text{tank}$$ $$\frac{x}{20}+\left(15-x\right)\left(\frac{1}{20}+\frac{1}{30}\right)=1\ $$ $$\frac{x}{20}+\frac{15-x}{20}+\frac{15-x}{30}=1\ \Leftrightarrow\ \ \frac{15}{20}+\frac{15-x}{30}=1$$ $$450+20\left(15-x\right)=600\ \Leftrightarrow\ \ 300-20x=150\ \Leftrightarrow\ 20x=150\ \Leftrightarrow\ x=7.5$$ This is why the correct answer is the option B.

I hope this answer may help you.

If you have a doubt, let me know.

Regards.

GMAT Prep From The Economist
We offer 70+ point score improvement money back guarantee.
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GMAT/MBA Expert

ErikaPrepScholar: Legendary Member; Posts: 503; Joined: Thu Jul 20, 2017 9:03 am; Thanked: 86 times; Followed by:15 members; GMAT Score:770

by ErikaPrepScholar » Thu Feb 08, 2018 8:13 am

For the first x minutes, the first tap works alone at a rate of 1 cistern/20 minutes. We can express the total amount filled in this time as:

$$x\ \min\left(\frac{1\ cistern}{20\ \min}\right)$$

For the remainder of the 15 minutes (in other words, for 15 - x minutes), the two taps work together at their respective rates. We can express the total amount filled in this time as:

$$\left(15\ -x\right)\ \min\left(\frac{1\ cistern}{20\ \min}+\frac{1\ cistern}{30\ \min}\right)$$

If we add the total amount filled in the first x minutes to the total amount filled in the remained of the 15 minutes, we should get one full cistern:

$$x\ \min\left(\frac{1\ cistern}{20\ \min}\right)+\left(15\ -x\right)\ \min\left(\frac{1\ cistern}{20\ \min}+\frac{1\ cistern}{30\ \min}\right)=1\ cistern$$

Or with our units cancelled:

$$x\left(\frac{1}{20}\right)+\left(15\ -x\right)\left(\frac{1}{20}+\frac{1}{30}\right)=1$$

Then we simplify to solve for x:

$$\frac{x}{20}+\frac{15-x}{20}+\frac{15-x}{30}=1$$ $$\frac{3x}{60}+\frac{45-3x}{60}+\frac{30-2x}{60}=1$$ $$3x+45-3x+30-2x=60$$ $$75-2x=60$$ $$2x=15$$ $$x=7.5$$

So the value of x is 7.5.

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https://gmat.prepscholar.com/gmat/s/

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GMAT/MBA Expert

Scott@TargetTestPrep: GMAT Instructor; Posts: 8086; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

by Scott@TargetTestPrep » Mon Jul 01, 2019 4:20 pm

BTGmoderatorLU wrote:Two taps can fill a cistern in 20 minutes and 30 minutes. The first tap was opened initially for x minutes after which the second tap was opened. If it took a total of 15 minutes for the tank to be filled, what is the value of x?

A. 5.0
B. 7.5
C. 9.0
D. 10.0
E. 12.5

We are given that the rate of the first cistern is 1/20 and the rate of the second cistern is 1/30. The first tap worked for 15 minutes, and the second tap worked for (15 - x) minutes. Thus:

(1/20)(15) + (1/30)(15 - x) = 1

3/4 + (15 - x)/30 = 1

Multiplying the entire equation by 60, we have:

45 + 30 - 2x = 60

2x = 15

x = 7.5

Answer: B

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Founder and CEO
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