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water pumps

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by LevelOne » Mon Jun 22, 2009 11:20 am
Two water pumps, working simultaneously at their respective constant rates, took exactly 4 hours to fill a certain swimming pool. If the constant rate of one pump was 1.5 times the constant rate of the other, how many hours would it have taken the faster pump to fill the pool if it had worked alone at its constant rate?

a. 5
b. 16/3
c. 11/2
d. 6
e. 20/3

thanks!
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by truplayer256 » Mon Jun 22, 2009 11:38 am
Let's call the two pumps Pump A and Pump B. Pump A has a rate of x and Pump B has a rate of 1.5(x). Using the formula:

1/Rate of Pump A to fill the pool + 1/ Rate of Pump B to fill the pool = 1/ Rate of Pump A and B together to fill the pool.

1/x+1/1.5(x)=1/4

5(x)/2=3(x^2)/8

3(x^2)/8-5x/2=0

x(3x/8-5/2)=0

x=0, x=20/3

Our answer should be 20/3 or E.
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by LevelOne » Mon Jun 22, 2009 11:53 am
thanks, the answer is E.

what exactly did you do to 1/x+1/1.5(x)=1/4 to arrive at
5(x)/2=3(x^2)/8 ?

I started off the same way but my solution is somehow different...

Also, how would you solve this one:

Six machines, each working at the same constant rate, together can complete a certain job in 12 days. How many additional machines, each working at the same constant rate, will be needed to complete the job in 8 days?

a. 2
b. 3
c. 4
d. 6
e. 8
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by truplayer256 » Mon Jun 22, 2009 12:05 pm
what exactly did you do to 1/x+1/1.5(x)=1/4 to arrive at
5(x)/2=3(x^2)/8 ?
well, I pretty much multiplied the 1.5(x) to the one and the x to the one to get 1.5(x)+x, and then multiplied the denominators together because we're adding two fractions. So then you get 2.5(x)/1.5x^(2)=1/4. Multiply both sides by 1.5x^2 and you get 2.5(x)=3x^2/2/(4)=5x/2=3x^2/8. Hope that helped.

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Six machines, each working at the same constant rate, together can complete a certain job in 12 days. How many additional machines, each working at the same constant rate, will be needed to complete the job in 8 days?
Since it takes 12 days for 6 machines to complete a certain job, it'll take 12*6 or 72 days for one machine to complete the job. Now we know that 1/9 of 72 is 8, right? so we need 9 machines to complete the job in 8 days. Since the question is asking you for how many additional machines are needed to complete the job in 8 days, the answer to this question will be 9-6=3 additional machines.
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by LevelOne » Mon Jun 22, 2009 12:10 pm
that's great! thanks
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Re: water pumps

by Stuart@KaplanGMAT » Mon Jun 22, 2009 1:36 pm
LevelOne wrote:Two water pumps, working simultaneously at their respective constant rates, took exactly 4 hours to fill a certain swimming pool. If the constant rate of one pump was 1.5 times the constant rate of the other, how many hours would it have taken the faster pump to fill the pool if it had worked alone at its constant rate?

a. 5
b. 16/3
c. 11/2
d. 6
e. 20/3

thanks!
I solved intuitively rather than using an equation.

I thought to myself, "Self, one pump is 1.5 times faster than the other. So, for every 1 unit of work done by slow pump, fast pump is doing 1.5 units of work. Or, put another way, that fast pump is doing 1.5/(1.5+1) of the work.

So Self, what does that mean? It means that fast pump is doing 1.5/2.5 or 3/5 of the work.

Self, myself is pondering, how does that help? Well, Self, if it's doing 3/5 of the work, then if it had to work all alone (which it might enjoy, given how slow that other pump is), then it's going to take 5/3 of the time to get the job done.

Aha! says myself, finally caught up on things... so I simply multiply:

4hours * 5/3 = 20/3 hours and select choice E!"
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by pops » Mon Jun 22, 2009 9:09 pm
aha...thats really cool !!
I was going by the method mentioned first above.. and it takes quite some time to solve the equation.. this way we can save lot time...
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