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

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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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02 Feb 2017, 03:53
1) A&B can do the job in $$\frac{6}{5}$$ of an hour; and their rate when working together is $$\frac{1}{6/5}=\frac{5}{6}$$
A&C can do the job in $$\frac{3}{2}$$ of an hour; and their rate when working together is $$\frac{1}{3/2}=\frac{2}{3}$$
B&C can do the job in 2 hours; and their rate when working together is $$\frac{1}{2}$$
2) If we combine all three rates $$\frac{5}{6}+\frac{2}{3}+\frac{1}{2}=\frac{5}{6}+\frac{4}{6}+\frac{3}{6}=\frac{12}{6}=2$$ (2 tanks in an hour) then we have each machine represented twice. If we divide it by 2 ($$\frac{2}{2}=1$$), then we see that pumps A, B, and C when working together can fill one tank in one hour
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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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08 Feb 2017, 10:20
chicagocubsrule wrote:
Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating simultaneously, can fill a certain tank in 6/5 hours; pumps A and C, operating simultaneously, can fill the tank in 3/2 hours; and 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. 1/4
D. 1
E. 5/6

We are given that pumps A and B, operating simultaneously, can fill a certain tank in 6/5 hours. Recall that rate = work/time. If we consider the work (filling the tank) as 1, then the combined rate of pumps A and B is 1/(6/5). We can express this in the following equation in which a = the time it takes pump A to fill the tank when working alone (thus 1/a is pump A’s rate) and b = the time it takes pump B to fill the tank when working alone (thus 1/b is pump B’s rate):

1/a + 1/b = 1/(6/5)

1/a + 1/b = 5/6

We are next given that pumps A and C, operating simultaneously, can fill the tank in 3/2 hours. We can create another equation in which c = the time it takes pump C to fill the tank alone.

1/a + 1/c = 1/(3/2)

1/a + 1/c = 2/3

Finally, we are given that pumps B and C, operating simultaneously, can fill the tank in 2 hours. We can create the following equation:

1/b + 1/c = ½

Next we can add all three equations together:

(1/a + 1/b = 5/6) + (1/a + 1/c = 2/3) + (1/b + 1/c = ½)

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

2/a + 2/b + 2/c = 5/6 + 4/6 + 3/6

2/a + 2/b + 2/c = 12/6

2/a + 2/b + 2/c = 2

Since we need to determine the combined rate of all three machines when filling 1 tank, we can multiply the above equation by ½:

(2/a + 2/b + 2/c = 2) x ½

1/a + 1/b + 1/c = 1

Since the combined rate of all 3 pumps is 1 and time = work/rate, the time needed to fill the tank when all 3 pumps are operating simultaneously is 1/1 = 1 hour.

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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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14 Aug 2017, 14:16
corrected myself

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

so resolving by b
b= 1/2 -c
a = 2/3 - c
then first equation a+b=5/6 is

1/2-c+2/3 - c = 5/6
2c + 7/6 = 5/6
2c = 2/6
c= 1/6

then find others by substituting c into the rest equations
a = 1/2 b=1/3 and c=1/6 by summing them we get 1
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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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25 Aug 2017, 08:44
When you look at the world do you just see one's and zero's 10101011 like the matrix? Solutions are ninja. Took me a while to figure this out.

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

a) 1/3
b) 1/2
c) 1/4
d) 1
e) 5/6

A and B = 5/6 --> 1/A+1/B=5/6
A and C = 2/3 --> 1/A+1/C=2/3
B and C = 1/2 --> 1/B+1/C=1/2

Q 1/A+1/B+1/C=?

Add the equations: 1/A+1/B+1/A+1/C+1/B+1/C=5/6+2/3+1/2=2 --> 2*(1/A+1/B+1/A+1/C)=2 --> 1/A+1/B+1/A+1/C=1

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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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21 Nov 2017, 07:50
$$\frac{1}{a} + \frac{1}{b} = \frac{5}{6}$$

$$\frac{1}{a} + \frac{1}{c} = \frac{2}{3}$$

$$\frac{1}{b} + \frac{1}{c} = \frac{1}{2}$$

When A, B and C are working together -

$$\frac{1}{a} + \frac{1}{b} + \frac{1}{a} + \frac{1}{c} + \frac{1}{b} + \frac{1}{c}$$ $$=$$ $$\frac{5}{6} + \frac{2}{3} + \frac{1}{2}$$

$$= \frac{36}{18}$$

= 2

2 ($$\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$$) = 2

$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$$ = 1

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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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23 May 2018, 07:25
Using the percentages approach, which makes the calculation very easy.

Let RA be rate of pump A

Let RB be rate of pump B

Lett RC be rate of pump C

RA and RB takes 6/5hours which means that there combined rate is 83% (reciprocal of 6/5)
Similarly RB and RC = 50% (Reciprocal of 2)
and RA + RC = 66%(Reciprocal of 3/2)

The reciprocal tell you the amount of work the pumps will do in one hour if working together.

So this gives us three very simple equations

RA+RB=83%
RB+RC=50%
RA+RC=66%

Solving them we get
RA=50%
RB=33%
and RC=16%

Since we need to find the total time required when all three are working, we add the above and we get approximately 100%

This means that it will take 1 hour to do the work if all three pumps are working together.

What is great about this method is we can skip a lot of fractional equations and hence save time.

Cheers!
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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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07 Oct 2018, 22:35
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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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04 Jan 2019, 06:38
chicagocubsrule wrote:
Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating simultaneously, can fill a certain tank in 6/5 hours; pumps A and C, operating simultaneously, can fill the tank in 3/2 hours; and 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. 1/4
D. 1
E. 5/6

2 (1/A + 1/B + 1/C) = 5/6 + 2/3 + 1/2 = 2
(1/A + 1/B + 1/C) = 2/2 = 1 Hence, OA = D
Re: Pumps A, B, and C operate at their respective constant rates. Pumps A &nbs [#permalink] 04 Jan 2019, 06:38

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