Pumps A, B and C operate at their respective constant rates

[MP]

Pumps A, B and C operate at their respective constant rates. Pumps A and B, simultaneously, can fill a certain tank in 6/5 hours. Pump A and C, operating simultaneously, can fill the tank in 3/2 hours; abd pumps B and C, operating simultaneously, can fill the tank in 2 hours. How many hours does it take pumps A, B, and C, operating simultaneously, to fill the tank?

(A) 1/3
(B) 1/2
(C) 2/3
(D) 5/6
(E) 1

Correct answer: (E)

I am unable to deduce the correct solution.

[RonPurewal]

Pumps A, B and C operate at their respective constant rates. Pumps A and B, simultaneously, can fill a certain tank in 6/5 hours. Pump A and C, operating simultaneously, can fill the tank in 3/2 hours; abd pumps B and C, operating simultaneously, can fill the tank in 2 hours. How many hours does it take pumps A, B, and C, operating simultaneously, to fill the tank?

(A) 1/3
(B) 1/2
(C) 2/3
(D) 5/6
(E) 1

Correct answer: (E)

I am unable to deduce the correct solution.

remember that rate = reciprocal of time taken to complete one job.
also, remember that rates are additive, so rate(pumps a AND b) = rate(pump a) + rate(pump b).
so:
rate(pumps a AND b) = 5/6
rate(pumps a AND c) = 2/3
rate(pumps b AND c) = 1/2

using the above fact about additive rates,
rate(pump a) + rate(pump b) = 5/6
rate(pump a) + rate(pump c) = 2/3
rate(pump b) + rate(pump c) = 1/2

you know you want the rate for all three pumps. from the symmetry of the above equations, it becomes apparent that we can find this by adding together all 3 equations:
2rate(pump a) + 2rate(pump b) + 2rate(pump c) = 5/6 + 2/3 + 1/2 = 2
rate(pump a) + rate(pump b) + rate(pump c) = 1
rate(pumps a AND b AND c) = 1 (because rates are additive)
time = reciprocal of 1 = 1

[MP]

Thanks a lot! I kept doing a lot of calculations and got things wrong.

I assumed that let A, B, C complete the job (indivisually) in a, b, c hours.

Hence their respective rates would be: 1/a, 1/b, 1/c

Then,
1/a + 1/b = 5/6
1/a + 1/c = 2/3
1/b + 1/c = 1/2

Then I was solving the 3 equations to get values of a, b, c. That consumed a lot of time and I had to guess the answer.

[rfernandez]

We're glad it makes sense now.

[gordon.thomas]

Answer should be (E) = 2 hours

In 1 hour, Pump A can empty 1/2 of the tank (it is already 1/2 full)
In 1 hour, Pump B can empty 1/3 of the tank

Since, pump A and B empty the tank together, and assume A alone empties all the water that is present (since it's half full), the problem reduces to how long can Pump C take to fill that much of the tank which B can empty. In 1 hour B can empty 1/3 of the tank.

In 1 hour, Pump C can fill 1/6 of the tank. Thus in 2 hours, C can fill 1/3 of the tank.

Hence in 2 hours, B removes everything that C fills and A removes whatever is present in the tank
[editor: this is apparently an answer to a completely different problem.]

[RonPurewal]

Answer should be (E) = 2 hours

In 1 hour, Pump A can empty 1/2 of the tank (it is already 1/2 full)
In 1 hour, Pump B can empty 1/3 of the tank

Since, pump A and B empty the tank together, and assume A alone empties all the water that is present (since it's half full), the problem reduces to how long can Pump C take to fill that much of the tank which B can empty. In 1 hour B can empty 1/3 of the tank.

In 1 hour, Pump C can fill 1/6 of the tank. Thus in 2 hours, C can fill 1/3 of the tank.

Hence in 2 hours, B removes everything that C fills and A removes whatever is present in the tank

what question are you answering?
certainly not the one in this thread.

[crissro]

I ran into this problem today and after reviewing all the postings I still have an issue.
The question is
"How many hours does it take pumps A, B, and C, operating simultaneously, to fill the tank?" and not "how much of the tank, the three pomps working simultaneously, can fill up in an hour?", so why my approach wasn't good?

A+B=1hr12min(6/5) 1 tank
A+C=1hr30min(3/2) 1 tank
B+C=2hr 1 tank
-----------------------------------
2(A+B+C)=4hr42min 3 tanks
A+B+C=2hr21min 3 tanks
A+B+C=47min 1 tank

I do not understand where is my mistake.
Thank you.

[jnelson0612]

I ran into this problem today and after reviewing all the postings I still have an issue.
The question is
"How many hours does it take pumps A, B, and C, operating simultaneously, to fill the tank?" and not "how much of the tank, the three pomps working simultaneously, can fill up in an hour?", so why my approach wasn't good?

A+B=1hr12min(6/5) 1 tank
A+C=1hr30min(3/2) 1 tank
B+C=2hr 1 tank
-----------------------------------
2(A+B+C)=4hr42min 3 tanks
A+B+C=2hr21min 3 tanks
A+B+C=47min 1 tank

I do not understand where is my mistake.
Thank you.

Hi crissro,
Here's your mistake: it's valid to add rates when two or more machines (or people) are working together. It does not work to add time together. Let me give you a simpler example:

Let's say that I can clean my house in 4 hours. My rate is then 1/4, the amount of my house that I can clean in one hour.

Now, let's say that I convince you to help me clean my house. :-) You are very speedy and can clean my house by yourself in 2 hours. Your rate is 1/2.

If we work together, in one hour I can clean 1/4 of the house and you can clean 1/2 of the house. Together we can clean 1/4+1/2 or 3/4 of the house. We add the rates to get our combined rate.

However, should we add our times to see how much time it takes us to clean the house? That would be 4 hours + 2 hours = 6 hours for us to clean the house. Does that make sense? No, because by working together we take *less* time to clean the house. In fact, it only takes us 4/3 of an hour to clean the house, using rate * time = work (3/4 *4/3 = 1).

Does this make more sense? To sum up, it is valid to add rates, but completely incorrect to add times in this kind of situation. The result you get will not be accurate or meaningful. This is why Ron recommends converting everything to rates right away by using reciprocals.

[crissro]

I ran into this problem today and after reviewing all the postings I still have an issue.
The question is
"How many hours does it take pumps A, B, and C, operating simultaneously, to fill the tank?" and not "how much of the tank, the three pomps working simultaneously, can fill up in an hour?", so why my approach wasn't good?

A+B=1hr12min(6/5) 1 tank
A+C=1hr30min(3/2) 1 tank
B+C=2hr 1 tank
-----------------------------------
2(A+B+C)=4hr42min 3 tanks
A+B+C=2hr21min 3 tanks
A+B+C=47min 1 tank

I do not understand where is my mistake.
Thank you.

Hi crissro,
Here's your mistake: it's valid to add rates when two or more machines (or people) are working together. It does not work to add time together. Let me give you a simpler example:

Let's say that I can clean my house in 4 hours. My rate is then 1/4, the amount of my house that I can clean in one hour.

Now, let's say that I convince you to help me clean my house. :-) You are very speedy and can clean my house by yourself in 2 hours. Your rate is 1/2.

If we work together, in one hour I can clean 1/4 of the house and you can clean 1/2 of the house. Together we can clean 1/4+1/2 or 3/4 of the house. We add the rates to get our combined rate.

However, should we add our times to see how much time it takes us to clean the house? That would be 4 hours + 2 hours = 6 hours for us to clean the house. Does that make sense? No, because by working together we take *less* time to clean the house. In fact, it only takes us 4/3 of an hour to clean the house, using rate * time = work (3/4 *4/3 = 1).

Does this make more sense? To sum up, it is valid to add rates, but completely incorrect to add times in this kind of situation. The result you get will not be accurate or meaningful. This is why Ron recommends converting everything to rates right away by using reciprocals.

Think you for taking the time to answer me.

[tim]

:)