SOLUTION: One pipe can empty a tank 2.5 faster than another pipe. Starting with a full tank, if both pipes are turned on, it takes 7.5 hours to empty the tank. How long does it take for the

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Question 1121249: One pipe can empty a tank 2.5 faster than another pipe. Starting with a full tank, if both pipes are turned on, it takes 7.5 hours to empty the tank. How long does it take for the faster pipe working alone to empty a full tank?
Found 4 solutions by josgarithmetic, ikleyn, Alan3354, greenestamps:
Answer by josgarithmetic(39621) About Me  (Show Source):
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x, time for the slower tank, alone
1%2Fx%2B2.5%2Fx=1%2F7.5

3.5%2Fx=1%2F7.5

x%2F3.5=7.5

x=%283.5%29%287.5%29

x=26.25
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Rate for the fast pipe, working alone,
2.5%2F26.25

TIME for the fast pipe,
26.25%2F2.5

highlight%2810.5%29 hours

Answer by ikleyn(52817) About Me  (Show Source):
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.
Let x be the rate of work of the slower pipe (measured in the tank volume per hour).


Then the rate of work of the faster pipe is 2.5x of the tank volume per hour.


The combined rate of the two pipes is then  x + 2.5x = 3.5x.


We are given


7.5*3.5*x = 1;  hence   x = 1%2F%287.5%2A3.5%29.


Then  2.5x = 2.5%2F%287.5%2A3.5%29 = 1%2F%283%2A3.5%29 = 1%2F10.5.


It means that the faster pipe will empty the tank in 10.5 hours, working alone.

Solved.


Answer by Alan3354(69443) About Me  (Show Source):
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2 answers, but...
2.5 times faster = 3.5 times as fast.

Answer by greenestamps(13203) About Me  (Show Source):
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Two tutors used similar methods to find an answer of 10.5 hours for the faster pipe to empty the tank working alone.

Then another tutor rightly pointed out that the answer of 10.5 hours can only be obtained by using the incorrect everyday interpretation of the phrase "2.5 times faster".

In sloppy everyday usage, "2.5 times FASTER THAN" is used to mean the same thing as "2.5 times AS FAST AS". But the two mean different things.

If x is the rate at which the slower pipe drains the pool, then if the faster pipe works 2.5 times AS FAST, then its rate of work is 2.5x.

But if the faster pipe works 2.5 times FASTER THAN the slower pipe, then its rate of work is x + 2.5x = 3.5x.

So, while the solution methods shown by the two tutors who got an answer of 10.5 hours are valid, they are not correct solutions to the problem as stated.

I will first show a very different method for getting the answer of 10.5 hours using the incorrect interpretation of the information given in the problem; then I will modify the answer using the correct interpretation.

Again let x be the rate at which the slower pipe drains the pool. If we use the interpretation that the faster pipe works 2.5 times AS FAST, then its rate is 2.5x.

The ratio of the two rates is x:2.5x, or 2x:5x, or 2:5.

That ratio means that, when the two pipes are working together, the slower pipe does 2/7 of the job and the faster pipe does 5/7 of the job.

Since the faster pipe does 5/7 of the job when working with the slower pipe, the time required for the faster pipe to drain the pool working alone will be 7/5 of the time required when the two are working together.

The problem tells us that the two pipes together take 7.5 hours; so the time required by the faster pipe alone would be 7/5 of 7.5 hours:

%287%2F5%29%2A7.5+=+%287%2F5%29%2815%2F2%29+=+21%2F2+=+10.5

The faster pipe alone would take 10.5 hours to drain the pool alone.

But, again, that is the answer to the wrong problem....

The actual rates are x and 3.5x; the faster drain does 7/9 of the total job; the time required for the faster tank to drain the pool alone using the correct interpretation of the given information is

%289%2F7%29%2A7.5+=+%289%2F7%29%2815%2F2%29+=+135%2F14

The correct answer to the problem as given is 135/14 hours.