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Manager  Joined: 02 Nov 2008
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Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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Question Stats: 71% (02:14) correct 29% (02:44) wrong based on 1801 sessions

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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

Originally posted by chicagocubsrule on 10 Nov 2009, 15:14.
Last edited by Bunuel on 17 Feb 2016, 05:57, edited 2 times in total.
Edited the question and added the OA
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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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58
57
chicagocubsrule wrote:
Thanks

Generally, if we are told that:
A hours is needed for worker A (pump A etc.) to complete the job --> the rate of A=$$\frac{1}{A}$$;
B hours is needed for worker B (pump B etc.) to complete the job --> the rate of B=$$\frac{1}{B}$$;
C hours is needed for worker C (pump C etc.) to complete the job --> the rate of C=$$\frac{1}{C}$$;

You can see that TIME to complete one job=Reciprocal of rate. eg 6 hours needed to complete one job (time) --> 1/6 of the job done in 1 hour (rate).

Time, rate and job in work problems are in the same relationship as time, speed (rate) and distance.

Time*Rate=Distance
Time*Rate=Job

Also note that we can easily sum the rates:
If we are told that A is completing one job in 2 hours and B in 3 hours, thus A's rate is 1/2 job/hour and B's rate is 1/3 job/hour. The rate of A and B working simultaneously would be 1/2+1/3=5/6 job/hours, which means that the will complete 5/6 job in hour working together.

Time needed for A and B working simultaneously to complete the job=$$\frac{A*B}{A+B}$$ hours, which is reciprocal of the sum of their respective rates. (General formula for calculating the time needed for two workers working simultaneously to complete one job).
Time needed for A and C working simultaneously to complete the job=$$\frac{A*C}{A+C}$$ hours.

Time needed for B and C working simultaneously to complete the job=$$\frac{B*C}{B+C}$$ hours.

General formula for calculating the time needed for THREE workers working simultaneously to complete one job is: $$\frac{A*B*C}{AB+AC+BC}$$ hours. Which is reciprocal of the sum of their respective rates: $$\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$$.

We have three equations and three unknowns:
1. $$\frac{1}{A}+\frac{1}{B}=\frac{5}{6}$$

2. $$\frac{1}{A}+\frac{1}{C}=\frac{2}{3}$$

3. $$\frac{1}{B}+\frac{1}{C}=\frac{1}{2}$$

Now the long way is just to calculate individually three unknowns A, B and C from three equations we have. But as we just need the reciprocal of the sum of relative rates of A, B and C, knowing the sum of $$\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\frac{AB+AC+BC}{ABC}$$ would be fine, we just take the reciprocal of it and bingo, it would be just the value we wanted.

If we sum the three equations we'll get:
$$2*\frac{1}{A}+2*\frac{1}{B}+2*\frac{1}{C}=\frac{5}{6}+\frac{2}{3}+\frac{1}{2}=2$$

$$\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=1$$, now we just need to take reciprocal of 1, which is 1.

So the time needed for A, B, and C working simultaneously to complete 1 job is 1 hour.

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

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72
21
You are adding the times, when you should be adding the rates.

Rates of:
A+B = 5/6 (tank per hour)
A+C = 2/3 (tank per hour)
B+C = 1/2 (tank per hour)

Combining these rates is like seeing "how much work would all of these pairs of machines do in 1 hour?"

(A+B)+(A+C)+(B+C) = 5/6 + 2/3 + 1/2 = 5/6 + 4/6 + 3/6 = 12/6 (tank per hour)

Therefore, 2A's, 2B's and 2C's working together would fill 2 tanks in an hour.
A single A, B, and C working together would fill 1 tank in 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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32
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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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Originally posted by Bunuel on 10 Nov 2009, 15:35.
Last edited by Bunuel on 10 Nov 2009, 15:49, edited 1 time in total.
VP  Joined: 05 Mar 2008
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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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Good technique..had another way but a few more unnecessary steps.
Manager  Joined: 10 Feb 2010
Posts: 130
Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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2
Ans e.

Let the rate of Pump A be a, Pump B b and Pump C c.

1/a + 1/b = 5/6

1/a + 1/c = 2/3

1/b + 1/c = 1/2

adding 2(1/a + 1/b + 1/c) = 5/6 + 2/3 + 1/2 = 5+4+3/6 = 2

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

or Pumps A,B and C take 1 hour to fill the tank.
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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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2
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

We knw that a+b are taking 6/5 hrs i.e. 1.2 hrs to fill a tank...
similarly B+C take 1.5 hrs and A+C take 2 hrs....

Adding all 3 equations we get

A+b + B+c + A+C = 1.2+1.5+2

2A+2B+2C = 4.7hrs
A+B+C = 2.35 hrs

can u please guide where am I going wrong???
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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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2
1
Work done by A & B together in one hour = 5/6

Work done by B & C together in one hour = 2/3

Work done by C & A together in one hour = 1/2

=> 2(Work done by A , B & C in one hour) = 5/6 + 2/3 + 1/2

=> Work done by A , B & C in one hour = 1/2 [ 5/6 + 2/3 + 1/2] = 1

So together all three were able to finish the work 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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1/A + 1/B = 5/6 portion of tank in 1 hr

1/A + 1/C = 2/3 portion of tank in 1 hr

1/B + 1/C = 1/2 portion of tank in 1 hr

2(1/A + 1/B + 1/C) = 5/6 + 2/3 + 1/2 = (5 + 4 + 3)/6 = 2

So (1/A + 1/B + 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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1
Bunuel wrote:
chicagocubsrule wrote:
Thanks

Generally, if we are told that:
A hours is needed for worker A (pump A etc.) to complete the job --> the rate of A=$$\frac{1}{A}$$;
B hours is needed for worker B (pump B etc.) to complete the job --> the rate of B=$$\frac{1}{B}$$;
C hours is needed for worker C (pump C etc.) to complete the job --> the rate of C=$$\frac{1}{C}$$;

You can see that TIME to complete one job=Reciprocal of rate. eg 6 hours needed to complete one job (time) --> 1/6 of the job done in 1 hour (rate).

Time, rate and job in work problems are in the same relationship as time, speed (rate) and distance.

Time*Rate=Distance
Time*Rate=Job

Also note that we can easily sum the rates:
If we are told that A is completing one job in 2 hours and B in 3 hours, thus A's rate is 1/2 job/hour and B's rate is 1/3 job/hour. The rate of A and B working simultaneously would be 1/2+1/3=5/6 job/hours, which means that the will complete 5/6 job in hour working together.

Time needed for A and B working simultaneously to complete the job=$$\frac{A*B}{A+B}$$ hours, which is reciprocal of the sum of their respective rates. (General formula for calculating the time needed for two workers working simultaneously to complete one job).
Time needed for A and C working simultaneously to complete the job=$$\frac{A*C}{A+C}$$ hours.

Time needed for B and C working simultaneously to complete the job=$$\frac{B*C}{B+C}$$ hours.

General formula for calculating the time needed for THREE workers working simultaneously to complete one job is: $$\frac{A*B*C}{AB+AC+BC}$$ hours. Which is reciprocal of the sum of their respective rates: $$\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$$.

We have three equations and three unknowns:
1. $$\frac{1}{A}+\frac{1}{B}=\frac{5}{6}$$

2. $$\frac{1}{A}+\frac{1}{C}=\frac{2}{3}$$

3. $$\frac{1}{B}+\frac{1}{C}=\frac{1}{2}$$

Now the long way is just to calculate individually three unknowns A, B and C from three equations we have. But as we just need the reciprocal of the sum of relative rates of A, B and C, knowing the sum of $$\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\frac{AB+AC+BC}{ABC}$$ would be fine, we just take the reciprocal of it and bingo, it would be just the value we wanted.

If we sum the three equations we'll get:
$$2*\frac{1}{A}+2*\frac{1}{B}+2*\frac{1}{C}=\frac{5}{6}+\frac{2}{3}+\frac{1}{2}=2$$

$$\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=1$$, now we just need to take reciprocal of 1, which is 1.

So the time needed for A, B, and C working simultaneously to complete 1 job is 1 hour.

Hope it helps.

Nice Explanation.

Thanks for clearing the concepts.
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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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Bunuel wrote:
chicagocubsrule wrote:
Thanks

Generally, if we are told that:
A hours is needed for worker A (pump A etc.) to complete the job --> the rate of A=$$\frac{1}{A}$$;
B hours is needed for worker B (pump B etc.) to complete the job --> the rate of B=$$\frac{1}{B}$$;
C hours is needed for worker C (pump C etc.) to complete the job --> the rate of C=$$\frac{1}{C}$$;

You can see that TIME to complete one job=Reciprocal of rate. eg 6 hours needed to complete one job (time) --> 1/6 of the job done in 1 hour (rate).

Time, rate and job in work problems are in the same relationship as time, speed (rate) and distance.

Time*Rate=Distance
Time*Rate=Job

Also note that we can easily sum the rates:
If we are told that A is completing one job in 2 hours and B in 3 hours, thus A's rate is 1/2 job/hour and B's rate is 1/3 job/hour. The rate of A and B working simultaneously would be 1/2+1/3=5/6 job/hours, which means that the will complete 5/6 job in hour working together.

Time needed for A and B working simultaneously to complete the job=$$\frac{A*B}{A+B}$$ hours, which is reciprocal of the sum of their respective rates. (General formula for calculating the time needed for two workers working simultaneously to complete one job).
Time needed for A and C working simultaneously to complete the job=$$\frac{A*C}{A+C}$$ hours.

Time needed for B and C working simultaneously to complete the job=$$\frac{B*C}{B+C}$$ hours.

General formula for calculating the time needed for THREE workers working simultaneously to complete one job is: $$\frac{A*B*C}{AB+AC+BC}$$ hours. Which is reciprocal of the sum of their respective rates: $$\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$$.

We have three equations and three unknowns:
1. $$\frac{1}{A}+\frac{1}{B}=\frac{5}{6}$$

2. $$\frac{1}{A}+\frac{1}{C}=\frac{2}{3}$$

3. $$\frac{1}{B}+\frac{1}{C}=\frac{1}{2}$$

Now the long way is just to calculate individually three unknowns A, B and C from three equations we have. But as we just need the reciprocal of the sum of relative rates of A, B and C, knowing the sum of $$\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\frac{AB+AC+BC}{ABC}$$ would be fine, we just take the reciprocal of it and bingo, it would be just the value we wanted.

If we sum the three equations we'll get:
$$2*\frac{1}{A}+2*\frac{1}{B}+2*\frac{1}{C}=\frac{5}{6}+\frac{2}{3}+\frac{1}{2}=2$$

$$\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=1$$, now we just need to take reciprocal of 1, which is 1.

So the time needed for A, B, and C working simultaneously to complete 1 job is 1 hour.

Hope it helps.

Serioulsy.. Nice explanation..
It will clear the basics for solving Time and Rate questions..
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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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Nice problem and nice solution. Sum all the given rates and divide by 2. Take the reciprocal of the result to get our required answer.
Thanks Bunuel for amazing explanations.
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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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1
This is an addition of rates problem

A+B = 5/6 (NOTICE the inversion here)
A+C = 2/3
B+C = 1/2

A+B+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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Given:

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

1/A +1/ B = 5/6

1/A + 1/C = 2/3

1/B + 1/C = 1/2

2(1/A + 1/B + 1/C) = 5/6 + 2/3 + 1/2 = (5+4+3)/6 = 12/6 = 2

1/A + 1/B + 1/C = 1

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

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$$\frac{1}{A}+\frac{1}{B}=\frac{5}{6}$$
$$\frac{1}{A}+\frac{1}{C}=\frac{2}{3}$$
$$\frac{1}{B}+\frac{1}{C}=\frac{1}{2}$$

Note: time= reciprocal of rate and rate= reciprocal of time

$$\frac{2}{A}+\frac{2}{B}+\frac{2}{C}=2$$
$$\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\frac{1}{1}$$

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

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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+ B = 6/5 = 1.2
A+C = 3/2 = 1.5
B+C = 2

Sum all 2 (A+B+C) =

formula for A+B+C = 1.2 * 1.5 * 2 / (1.5 * 2 + 1.2 * 2 + 1.5 * 1.2) = 1/2.

For 2 (A+B+C) = 2 *1/2 = 1
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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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1
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

Rate of A + B = 5/6
Rate of A + C = 2/3
Rate of B + C = 1/2

therefore
2(A + B + C) = 5/6 + 2/3 + 1/2
2(A + B + C) = 12/6
rate of A + B + C = 1

time is 1/rate = 1 hr
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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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1
fastest approach...
A+B+A+C+B+C = 2(A+B+C)
this is equal to 5/6 +2/3 + 1/2 or 12/6 = 2.
divide by 2
A+B+C will fill in 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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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

Bunuel -

By this same logic, Why am I not able to solve by adding the Times of each of the combined machines and solving for combined rate (then taking the reciprocal to solve for time)?

(A+B+A+C+B+C) = 6/5 + 3/2 + 2/1......2(A+B+C)=47/10

(A+B+C) = 47/20

With 3 full tanks done:

Rate = 47/20 / 3

Rate = 47/60

Time = 60/47

I see this is not an answer choice, but why is my logic wrong here?
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Re: Pumps A, B, and C operate at their respective constant rates. Pumps A  [#permalink]

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KFBR392 wrote:
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

Bunuel -

By this same logic, Why am I not able to solve by adding the Times of each of the combined machines and solving for combined rate (then taking the reciprocal to solve for time)?

(A+B+A+C+B+C) = 6/5 + 3/2 + 2/1......2(A+B+C)=47/10

(A+B+C) = 47/20

With 3 full tanks done:

Rate = 47/20 / 3

Rate = 47/60

Time = 60/47

I see this is not an answer choice, but why is my logic wrong here?

You CAN sum the rates but you CANNOT sum the times.
_________________ Re: Pumps A, B, and C operate at their respective constant rates. Pumps A   [#permalink] 04 Nov 2016, 08:37

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