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# Machine A can complete a certain job in x hours. Machine B

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Machine A can complete a certain job in x hours. Machine B [#permalink]  27 Jan 2012, 15:46
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Machine A can complete a certain job in x hours. Machine B can complete the same job in y hours. If A and B work together at their respective rates to complete the job, which of the following represents the fraction of the job that B will not have to complete because of A's help?

A. (x – y)/(x + y)
B. x/(y–x)
C. (x+y)/xy
D. y/(x-y)
E. y/(x+y)
[Reveal] Spoiler: OA
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Re: Work Prob - Part of the Job done [#permalink]  27 Jan 2012, 16:20
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docabuzar wrote:
Machine A can complete a certain job in x hours. Machine B can complete the same job in y hours. If A and B work together at their respective rates to complete the job, which of the following represents the fraction of the job that B will not have to complete because of A's help?

A. (x – y)/(x + y)
B. x/(y–x)
C. (x+y)/xy
D. y/(x-y)
E. y/(x+y)

Working together A and B complete the job in $$\frac{xy}{x+y}$$ hours (as time is a reciprocal of rate then take the reciprocal of combined rate, which is $$\frac{1}{x}+\frac{1}{y}=\frac{x+y}{xy}$$);

Now, in $$\frac{xy}{x+y}$$ hours A will complete $$\frac{1}{x}*\frac{xy}{x+y}=\frac{y}{x+y}$$ part of the job (rate*time=job) and this will be the part which B will not have to complete.

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Re: Machine A can complete a certain job in x hours. Machine B [#permalink]  24 Jun 2012, 18:55
Hi Bunuel,

Can you help me understand that when we want to calculate the work just done by A and not B of the total work which in this case we assume it as 1. Then why do we multiply the rate of only A with the total time?

I can't get my head around this concept. Is it one of those things that I just accept and move on.

Regards.
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Re: Machine A can complete a certain job in x hours. Machine B [#permalink]  24 Jun 2012, 21:11
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Take an example to understand it.
A takes 2 hrs (= x) to complete a work alone and B takes 2 hrs (= y) to complete a work alone. Together, they will complete the work in 1 hr = [xy/(x+y) = 2*2/(2+2)]
So A does 1/2 of the work in 1 hr (A's rate of work) and B does 1/2 of the work in 1 hr (B's rate of work).

So when they were working together, each needed to work for only 1 hr.
How much work was done by A in that one hr? 1*(1/2) = (1/2)
So because of A's help, B doesn't need to do half of the work i.e. work done by A.
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Re: Machine A can complete a certain job in x hours. Machine B [#permalink]  19 Sep 2012, 16:21
My solution might be very simple:

If we pick numbers like x = 3hrs and y = 4hrs then it takes 7hrs to complete two units for both of them. (We can use two units because we are looking for fractional work, not a total quantity #). If it takes 7hrs between the two of them then Ma does 3/7 or the work and Mb does 4/7 of the work. This means that Mb is not doing 3/7 of the work and you can plug in your answer choices to find out which one gives you x=3, y=4 -> 3/7
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Re: Machine A can complete a certain job in x hours. Machine B [#permalink]  15 Nov 2012, 00:45
VeritasPrepKarishma wrote:
Take an example to understand it.
A takes 2 hrs (= x) to complete a work alone and B takes 2 hrs (= y) to complete a work alone. Together, they will complete the work in 1 hr = [xy/(x+y) = 2*2/(2+2)]
So A does 1/2 of the work in 1 hr (A's rate of work) and B does 1/2 of the work in 1 hr (B's rate of work).

So when they were working together, each needed to work for only 1 hr.
How much work was done by A in that one hr? 1*(1/2) = (1/2)
So because of A's help, B doesn't need to do half of the work i.e. work done by A.

This explanation is vague since you picked 1/2. We cannot see how it works because it does not matter what you do with x and y it's around 1/2.
I see that formula is absolutely correct but don't understand why if the value of 1/x is 2/6 and 1/y is 3/6, the portion which y should not do comes out as 3/5 not 2/5.
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Re: Machine A can complete a certain job in x hours. Machine B [#permalink]  15 Nov 2012, 01:25
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$$\frac{1}{x}+\frac{1}{y}=W$$

Combining their efforts they could finish: $$W=\frac{x+y}{xy}$$

This means it will take them t to complete the job.

$$\frac{x+y}{xy}(t)=1==>t=\frac{xy}{x+y}$$

The job done by x will be the job that y doesn't have to worry about.

$$W=\frac{1}{x}(xy/x+y)==>W=\frac{y}{x+y}$$
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Re: Machine A can complete a certain job in x hours. Machine B [#permalink]  15 Nov 2012, 02:03
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felixjkz wrote:

This explanation is vague since you picked 1/2. We cannot see how it works because it does not matter what you do with x and y it's around 1/2.
I see that formula is absolutely correct but don't understand why if the value of 1/x is 2/6 and 1/y is 3/6, the portion which y should not do comes out as 3/5 not 2/5.

The explanation was meant for saswani who had trouble understanding "Then why do we multiply the rate of only A with the total time?"
It doesn't matter what the numbers are - as long as they are easy to work with - you can make out what's going on.

As for your question "why if the value of 1/x is 2/6 and 1/y is 3/6, the portion which y should not do comes out as 3/5 not 2/5."

If x = 3 and y = 2,
A does 1/3rd work in 1 hr and B does 1/2 work in 1 hr.
When they work together, they complete the work in 1/(1/2 + 1/3) = 6/5 hrs

In 6/5 hrs, A does (1/3)*(6/5) = 2/5 work
and B does (1/2)*(6/5) = 3/5 work

So B does NOT need to do 2/5 of the work. Answer still (E).
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Re: Machine A can complete a certain job in x hours. Machine B [#permalink]  07 Jan 2013, 03:09
Can someone please explain the formula xy / x+y to state the work done together. I thought I had a well understanding of this concept, however when I saw this question, I lost it completely. Thanks in advance
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Re: Machine A can complete a certain job in x hours. Machine B [#permalink]  07 Jan 2013, 03:22
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The amount of work that A can do in 1 hour ( or the rate at which A works ) is 1/x, since A takes x hours to complete the whole work. Similarly the amount of work that B can do in 1 hour is 1/y. Now, total work that they can do, when working together, in one hour is 1/x + 1/y.
You can also write it as 1/W = 1/x + 1/y = (x + y ) / xy. 1/W is the rate at which they are working.
Hence, to complete the total work they need xy / (x + y) hours.
Please let me know if anything is not clear.
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Re: Machine A can complete a certain job in x hours. Machine B [#permalink]  07 Jan 2013, 03:25
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KevinBrink wrote:
Can someone please explain the formula xy / x+y to state the work done together. I thought I had a well understanding of this concept, however when I saw this question, I lost it completely. Thanks in advance

Check here: two-consultants-can-type-up-a-report-126155.html#p1030079 and here: work-word-problems-made-easy-87357.html

Hope it helps.
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Re: Machine A can complete a certain job in x hours. Machine B [#permalink]  07 Jan 2013, 03:35
Can someone explain this step a little bit further 1/W = 1/x + 1/y = (x + y ) / xy?
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Re: Machine A can complete a certain job in x hours. Machine B [#permalink]  07 Jan 2013, 03:57
Iam really confused right now, I understand everything completely except for this essential part. I went through the stuff you told me Bunuel and I understand everything stated in there. But how do you come from 1/x + 1/y --> (x+y) / xy??? Thanks in advance
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Re: Machine A can complete a certain job in x hours. Machine B [#permalink]  07 Jan 2013, 04:10
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KevinBrink wrote:
Iam really confused right now, I understand everything completely except for this essential part. I went through the stuff you told me Bunuel and I understand everything stated in there. But how do you come from 1/x + 1/y --> (x+y) / xy??? Thanks in advance

You mean you don't get how 1/x + 1/y equals to (x+y)/xy?

The same way as $$\frac{1}{2}+\frac{1}{3}=\frac{3+2}{2*3}=\frac{5}{6}$$.
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Re: Machine A can complete a certain job in x hours. Machine B [#permalink]  07 Jan 2013, 04:14
Thanks, thanks, thanks!!!! Iam just not that good at re-rewriting mathematical stuff with variables. But with the numerical example I understand it completely. Thank your very much!
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Re: Machine A can complete a certain job in x hours. Machine B [#permalink]  16 Apr 2014, 01:26
(1/x) of the work is completed by Machine A in one hour.
(1/y) of the work is completed by Machine B in one hour.
Combined rate(one hour) is 1/x + 1/y = (x+y/xy).
fraction of the job that B will not have to complete because of A's help?
(1/x) of the work will be completed by A.
As we are asked about fraction , (1/x) / (x+y/xy) = (y/x+y).
Hence E
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Re: Machine A can complete a certain job in x hours. Machine B [#permalink]  13 Jan 2015, 01:35
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.......................... Rate ..................... Time ................. Work Done

Machine A ............... $$\frac{1}{x}$$ ...................... x ........................ 1 (Assumption)

Machine B ................ $$\frac{1}{y}$$ ....................... y ....................... 1

Combined rate ............ $$\frac{1}{x} + \frac{1}{y} = \frac{x+y}{xy}$$ ............. $$\frac{xy}{x+y}$$ ................... 1

Work done due to A's help = Rate of A * Time required for the combined

$$= \frac{1}{x} * \frac{xy}{x+y} = \frac{y}{x+y}$$

This will be the benefit for B

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Re: Machine A can complete a certain job in x hours. Machine B   [#permalink] 13 Jan 2015, 01:35
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