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Three hoses work to fill a tub at a different rate. Hose

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Three hoses work to fill a tub at a different rate. Hose

by Gmat_mission » Sat Nov 23, 2019 3:11 am
Three hoses work to fill a tub at a different rate. Hose A and B, working together, can fill the tub in 6/5 of an hour. Hoses A and C can fill it in 3/2 an hour. Houses B and C can fill it in 2 hours. How long does it take all 3 hoses, working together, to fill the tub?

A. 3/10

B. 2/5

C. 1/2

D. 1

E. 6/5

[spoiler]OA=D[/spoiler]

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by Brent@GMATPrepNow » Sat Nov 23, 2019 7:35 am
Gmat_mission wrote:Three hoses work to fill a tub at a different rate. Hose A and B, working together, can fill the tub in 6/5 of an hour. Hoses A and C can fill it in 3/2 an hour. Hoses B and C can fill it in 2 hours. How long does it take all 3 hoses, working together, to fill the tub?

A. 3/10
B. 2/5
C. 1/2
D. 1
E. 6/5
-------ASIDE----------------------
For work questions, there are two useful rules:

Rule #1: If a person can complete an entire job in k hours, then in one hour, the person can complete 1/k of the job
Example: If it takes Sue 5 hours to complete a job, then in one hour, she can complete 1/5 of the job. In other words, her work rate is 1/5 of the job per hour

Rule #2: If a person completes a/b of the job in one hour, then it will take b/a hours to complete the entire job
Example: If Sam can complete 1/8 of the job in one hour, then it will take him 8/1 hours to complete the job.
Likewise, if Joe can complete 2/3 of the job in one hour, then it will take him 3/2 hours to complete the job.
----------------------------------------------
Let's use the above rules to solve the question. . . .

Hoses A and B, working together, can fill the tub in 6/5 of an hour
Applying Rule #1, we can conclude that A and B COMBINED can fill 1/(6/5) of the tub in ONE hour
In other words, the combined rate of A and B = 5/6 tubs per HOUR

Hoses A and C can fill it in 3/2 an hour
Applying Rule #1, we can conclude that A and B COMBINED can fill 1/(3/2) of the tub in ONE hour
In other words, the combined rate of A and C = 2/3 tubs per HOUR

Hoses B and C can fill it in 2 hours.
So, the combined rate of B and C = 1/2 tubs per HOUR

Let A, B, C be the RATES for each hose.
So, we can write:
A + B = 5/6
A + C = 2/3
B + C = 1/2

Add ALL three equations to get:
2A + 2B + 2C = 5/6 + 2/3 + 1/2
= 5/6 + 4/6 + 3/6
= 12/6
= 2

This means that, the COMBINED rate of all 6 (2 A's, 2 B's and 2 C's) hoses is 2 tubs PER HOUR
This means that, the COMBINED rate of 1 A, 1 B and 1 C is 1 tub PER HOUR

So if all three pools work together they can fill the tub in one hour

Answer: D

Cheers.
Brent
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by deloitte247 » Sat Nov 30, 2019 12:31 pm
$$Work\ rate=\frac{Workdone}{Time}$$
$$WorkRate\ of\ Hose\ A+B=\frac{1\ tub}{\frac{6}{5}}=\frac{5}{6}$$
$$WorkRate\ of\ Hose\ A+C=\frac{1\ tub}{\frac{3}{2}}=\frac{2}{3}$$
$$WorkRate\ of\ Hose\ B+C=\frac{1\ tub}{2}=\frac{1}{2}$$
A + B = 5/6 ------ eqn. (1)
A + C = 2/3 ------ eqn. (2)
B + C = 1/2 ------ eqn. (3)
From eqn. (1), A = 5/6 - B
From eqn. (2), C = 2/3 - 5/6 + B
$$C=\frac{4-5}{6}+B\ \ \ \ \ \ =-\frac{1}{6}+B$$
From eqn. (3), B + C = 1/2; where C = -1/6 + B
$$B+\left(\frac{-1}{6}+B\right)=\frac{1}{2}$$
$$2B=\frac{1}{2}+\frac{1}{6}$$
$$B=\frac{\left(\frac{4}{6}\right)}{2}=\frac{4}{12}=\frac{1}{3}$$
From eqn. (3), B + C = 1/2; where B = 1/3
1/3 + C = 1/2
$$C=\frac{1}{2}-\frac{1}{3}\ \ \ \ \ \ =\ \frac{3-2}{6}\ \ \ =\frac{1}{6}$$
From eqn. (2), A+C = 2/3 where C = 1/6
$$A=\frac{2}{3}-\frac{1}{6}=\frac{4-1}{6}\ \ \ \ \ =\frac{1}{2}$$
How long does it take all the 3 hoses working together to fill the tub?
$$Rate\ of\ A+B+C=\frac{Workdone}{Time}$$
$$Workdone=1\ tub$$
$$Rate\ of\ A+B+C=\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=\ \ \ \frac{6}{6}\ \ \ =\ 1$$
$$So,\ time=\frac{workdone}{rate}=\frac{1}{1}=1\ hour$$
$$Correct\ answer\ =\ option\ D$$

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hi

by Scott@TargetTestPrep » Sun Dec 08, 2019 7:13 pm
Gmat_mission wrote:Three hoses work to fill a tub at a different rate. Hose A and B, working together, can fill the tub in 6/5 of an hour. Hoses A and C can fill it in 3/2 an hour. Houses B and C can fill it in 2 hours. How long does it take all 3 hoses, working together, to fill the tub?

A. 3/10

B. 2/5

C. 1/2

D. 1

E. 6/5

[spoiler]OA=D[/spoiler]

Source: GMAT Prep
We can create the following equations:

1/A + 1/B = 1/(6/5) = â…š

1/A + 1/C = 1/(3/2) = 2/3

1/B + 1/C = ½

Summing the 3 equations, we have:

2/A + 2/B + 2/C = 5/6 + 2/3 + 1/2 = 5/6 + 4/6 + 3/6 = 12/6 = 2

Dividing the equation by 2, we have:

1/A + 1/B + 1/C = 1

The combined rate for the 3 machines is 1. Thus, it takes one hour for all 3 machines working together to fill the tub.

Answer: D

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Founder and CEO
[email protected]

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by [email protected] » Thu Dec 12, 2019 11:45 am
Hi All,

We're told that three hoses work to fill a tub at a different rate. Hose A and B, working together, can fill the tub in 6/5 of an hour, Hoses A and C can fill it in 3/2 an hour and Houses B and C can fill it in 2 hours. We're asked how long it would take all 3 hoses, working together, to fill the tub. The formal 'math' approach to this question is long-winded and step-heavy; thankfully, the answer choices are sufficiently 'spread out' that you can answer this question without doing too much math at all (just a bit of logic and some comparisons will get you the correct answer).

To start, the 'fastest pair' of Hoses is clearly A and B (since together they can fill the tub in 6/5 of an hour). Adding an additional Hose (re: Hose C) will make the work go faster, so the correct answer MUST be less than 6/5. The question then becomes "how fast" is Hose C... ? We can think about that in terms of a comparison....

Hoses A and B can fill the tub in 6/5 of an hour
Hoses B and C can fill the tub in 2 hours

Clearly, when we replace Hose A with Hose C, filling the tub takes a LOT longer (66 2/3% LONGER!), so Hose C is NOT a 'fast' Hose. Compare Answer C (1/2 an hour) to Answer E (6/5 of an hour) and you'll see that Answer C is MORE THAN DOUBLE the speed of Answer E. Hose C could not possibly speed up the work that much - and Answers A and B would mean that Hose C was even faster than that. Thus, the only Answer that makes any sense is D.

Final Answer: D

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