Machine A can process 6000 envelopes in 3 hours. Machines B

I think you did everything right.

Let the time needed for A, B and C working individually to process 6,000 envelopes be \(a\), \(b\) and \(c\) respectively.

Now, as "A can process 6,000 envelopes in 3 hours" then \(a=3\);

As "B and C working together but independently can process the same number (6,000) of envelopes in 2.5 hours" then \(\frac{1}{b}+\frac{1}{c}=\frac{1}{2.5}=\frac{2}{5}\);

Also, as "A and C working together but independently process 3000 envelopes in 1 hour", then A and C working together but independently process 2*3,000=6,000 envelopes in 2*1=2 hours: \(\frac{1}{a}+\frac{1}{c}=\frac{1}{2}\) --> as \(a=3\) then \(c=6\);

So, \(\frac{1}{b}+\frac{1}{6}=\frac{2}{5}\) --> \(b=\frac{30}{7}\), which means that B produces 6,000 envelopes in 30/7 hours, thus it produces 12,000 envelopes in 60/7 hours.

Answer: E.

Adding to Murali's approach....

A = 2000
B+C = 2400
A+C = 3000 => C = 3000-1 = 3000-2000 = 1000

=> B = 2400 -C =2400-1000 =1400

hence B can process 1400 Envelopes in 1hour...how much time wud it take B to process 12000 Envelopes = 12000/1400 = 60/7

For 1 hour-

Machine A rate- 2000 envelopes
Machine B+C rate- 2400 envelopes
Since A + C = 3000 envelopes A's rate is 2000 envelopes as above, C has a rate of 1000 envelopes per hour.
Which makes machine B's rate as 1400 envelopes per hour.

Thus, it will take 8 hours to manufacture 12000 envelopes.

Answer:-E

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Machine A takes 3 hours for 6000 envelopes. Thus, Machine A would take exactly 6 hours for 12000 envelopes. Also, we know that machines B and C, working together, can produce the same no of envelopes in 2.5 hours. Thus, if\(r_B\) and\(r_C\) are the rates respectively , we know that\((r_B+r_C)*\frac{5}{2}\) = 6000 --> \((r_B+r_C) = 2400\). Thus, even if we assume that \(r_B\) = 2000 (which is the same rate as that of Machine A), Machine B would again need 6 hours. However, as\(r_C\)= 1000, we know for sure that\(r_B\) <2000. Thus, the only option more than 6 hours is E(Assuming that the correct OA is provided with the question).

I did this but shouldn't the work take 9 hours instead?
In 8 hours machine B would have made 1400 * 8 = 11200 envelopes.
In order to make 12000 it would require a fraction of an hour to create 200 more envelopes.
Am I mistaken?

As mentioned above, the OA is incorrect. In fact, the options are incorrect since none of them is 60/7 hrs (which is the answer).

Edited the options.

Check for a solution here: machine-a-can-process-6000-envelopes-in-3-hours-machines-b-105362.html#p823509 or here: machine-a-can-process-6000-envelopes-in-3-hours-machines-b-105362.html#p823655

How much time should one take in solving these kind of questions which involves though simple yet a lot of calculations?

This can be easily done in under 2 mins. If you look at the explanation provided above:


To make the question straight forward, first thing you can do is make the work the same for all:
Machine A processes 12000 envelopes in - 6 hrs
Machines B and C process 12000 in - 5 hrs
Machines A and C process 12000 in - 4 hrs
I chose to get them all to 12000 since my question has 12000 in it. Also, I easily get rid of all decimals.

Almost no calculations till here

Now, I just find 1/6 + 1/c = 1/4 and get c = 12
and 1/b + 1/12 = 1/5 so b = 60/7 hrs

You should be comfortable with manipulating fractions.
1/c = 1/4 - 1/6 = 2/24 = 1/12
So c = 12 (Finding c should take just a few seconds)

You can either take the amount of work done as the same as Karishma has done or take the work done by each in the same time. I will do the latter

1. Work done in 1 hr by A is 2000 envelopes
2. Work done in 1 hr by A and C is 3000 envelopes
3. So work done in 1 hr by C is 1000 envelopes
4. Work done in 1 hr by B and C is 2400 envelopes
5. So work done in 1 hr by B is 1400 envelopes
6. So to process 12000 envelopes B will take 12000/1400 hrs = 60/7 hrs

So the answer is choice E

Is this a tricky way of saying "working together"? I mean, can I treat that phrase just like how you would when you combine the rates of machine A and machine C? Is there a question where GMAT asks two or more entities working together depending on each other?

Yes. This means that they work independently of each other but they are turned on simultaneously.

1/A = 2000

1/B+1/C = 6000/2.5=2400 ---------(1)
1/A+ 1/C = 3000 -----------------------(2)

(1) - (2)
1/B-1/A =2400-3000
> 1/B - 2000 = 2400-3000
> 1/B = 1400

Rate = 1400
Work= 12000

Time= 12000/1400 =60/7

Feed me kudos if it is helpful for you :D

This is not the original question. In the original question (from Kaplan) the time given for mashines B and C working together is 2\frac{2}{5} and the correct answer choice is E, which equals to 8 and not \(\frac{60}{7}\)

Machine A can process 6000 envelopes in 3 hours. Machines B and C working together but independently can process the same number of envelopes in 2.5 hours. If Machines A and C working together but independently process 3000 envelopes in 1 hour, then how many hours would it take Machine B to process 12000 envelopes.

A. 2
B. 3
C. 4
D. 6
E. 60/7

let a,b,c=respective rates
a=2000
(a+c)-(b+c)=a-b=600
b=1400
12000/1400=60/7 hours
E

Instead of 2.5 hours , 2 hours and (2/5) mins then answer will be 4 hours . Correct?

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