Work of Pumps problem

[mcheytan]

Two pumps can fill a swimming pool together, working at same constant rates in 4 hours. If one of the pumps is working 1.5times the rate of the other, how long it needs to fill the swimming pool alone.
Ok, I have P1=P2, in the beginning. They total 4 hours, therefore, I can find P1 and P2=x
1/P1+1/P2=1/t, t=4h, therefore P1=P2=8
One of the pumps has 1.5 times the rate of the other, therefore P1=1.5P2, we do the equation as follows: 1.5/8=1/t2, where t2 is the time needed for the faster pump to fill the pool. t=16/3.

The answer I am given is 20/3, and this answer I get when I do the equation: 1/x+1/(1.5x)=1/4=>x=20/3, but this equation is not correct.....
Can you help me what to do?

 

[Deleted member 4993]

Two pumps can fill a swimming pool together, working at same constant rates in 4 hours. If one of the pumps is working 1.5times the rate of the other, how long it needs to fill the swimming pool alone.
Ok, I have P1=P2, in the beginning. They total 4 hours, therefore, I can find P1 and P2=x
1/P1+1/P2=1/t, t=4h, therefore P1=P2=8
One of the pumps has 1.5 times the rate of the other, therefore P1=1.5P2, we do the equation as follows: 1.5/8=1/t2, where t2 is the time needed for the faster pump to fill the pool. t=16/3.

The answer I am given is 20/3, and this answer I get when I do the equation: 1/x+1/(1.5x)=1/4=>x=20/3, but this equation is not correct.. ...<<<< how do you know that
Can you help me what to do?

Assume

Pump 1 fills the pool in t hours

Pump 1 fills the pool in 1/t fraction of the pool in 1hour ..........................(1)

Pump 2 fills the pool in 1.5 * t hours

Pump 1 fills the pool in 1/(1.5 * t) fraction of the pool in 1hour..........................(2)


Pump (1+2) fills the pool in 4 hours

Pump (1+2) fills the pool in 1/4 fraction of the pool in 1hour..........................(3)

Combining (1), (2) & (3), we get

1/t + 1/(1.5*t) = 1/4

 

[mcheytan]

Yes but the pumps are working together at constant rates which are the same for two of them, that is how they need 4 hours to complete the task.
that is why each of their work for 1/x, and together they do 1/x+1/x=1/4. Both of them together, at same constant rates fill the pool for 4 hours..
the 1.5 times x comes later, when only one of the pumps is working, and x is the same as in the above equation.

 

[Denis]

You seem to be pumping in circles, mcheytan !

Make a simple example for yourself, to SEE how it works.

Say you have a 1200 gallon tank and A can put in 40 gallons per hour, B can put in 1.5 of A = 60 gallons per hour:
so together they put in 100 gallons per hour, so it takes 1200/100 = 12 hours, or 1/12 of job in 1 hour; understand? ***

A working alone takes 1200/40 = 30 hours; so does 1/30 of job in 1 hour ****
B working alone takes 1200/60 = 20 hours; so does 1/20 of job in 1 hour

A + 3A/2 = 1/12 *** ; solve for A:
A = 1/30 **** see it?

Now do your problem same way...

 

[mcheytan]

I understand how you get the work done in one hour. I just do not understand why we add up together the work for one hour for the two machines (1/x + 3/2x) and put it equal to 1/4, which is the total time for the two machines if they work together at the SAME speed, not when one is 1.5 the other?????


OK, if each pump was working individually at their old constant rate, each will need 8 hours to fill the pool. Since they work together at same speeds, they do it for 4 (half of 8h).

Shouldn't we make 1/1.5x=1/t, where t is the unknown time for the pump when she works 1.5 times the regular work which is 8 hours???
Then we have 3/16=1/t???

Oh...I feel stupid...I just want to logically understand it...

 

[stapel]

I understand how you get the work done in one hour. I just do not understand why we add up together the work for one hour for the two machines (1/x + 3/2x)...

You add their hourly contributions because they are working together (at the same time) and you are assuming that their labors are additive.

This is not always the case in real life, of course, the classic counter-example being "nine women, one month, one baby". :wink:

...and put it equal to 1/4, which is the total time for the two machines if they work together at the SAME speed...

No; this is how long they take working together. Oh, and, by the way, each pump works at a constant rate throughout. (That is, the two pumps don't share the same rate; each pump works consistently at its own rate, which remains the same until the task is completed.)

Granted, the wording could have been a bit more clear...

Eliz.