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Running at their respective constant rate, machine X takes 2 days long

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Running at their respective constant rate, machine X takes 2 days long [#permalink] New post 05 Nov 2004, 23:42
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Running at their respective constant rate, machine X takes 2 days longer to produce w widgets than machines Y. AT these rates, if the two machines together produce 5w/4 widgets in 3 days, how many days would it take machine X alone to produce 2w widgets.

A. 4
B. 6
C. 8
D. 10
E. 12

OPEN DISCUSSION OF THIS QUESTION IS HERE: running-at-their-respective-constant-rate-machine-x-takes-98599.html
[Reveal] Spoiler: OA
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Re: Running at their respective constant rate, machine X takes 2 days long [#permalink] New post 06 Nov 2004, 00:06
I am getting 12. E. Hope havent done any calculation errors..

approach..
let y=no. of days taken by Y to do w widgets. Then X will take y+2 days.

1/(y+2) +1/y = 5/12 (5/12 is because (5/4)w widgets are done in 3 days. So, x widgets will be done in 12/5 days or 5/12 th of a widget in a day)

Solving, we have y = 4

=>X takes 6 days to doing x widgets. So, he will take 12 days to doing 2w widgets.
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Re: Running at their respective constant rate, machine X takes 2 days long [#permalink] New post 06 Nov 2004, 04:32
I also get 12 (E).

My approach was a little more complicated.

Work formula : COMBINED TIME =(A*B) : (A+B) , where A- time for 1st machine , B - time for 2nd machine.

(2+z) days will take machine A to produce 1 widg. and
z days will take machine to produce 1 widg.

Thus, (z*(z+2)) : (z+z+2)*5/4 =3 (days)

5z^2 +10z - 24 =0

z=4 ; z=-1,2 ( negative is not appropriate so we rid of it )

Consequently,machine A takes (2+z)days = 2+4= 6 (days to produce one widg.); and (6*2) = 12 (days to produce two widg.).

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Re: Running at their respective constant rate, machine X takes 2 days long [#permalink] New post 23 Jan 2015, 08:48
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Re: Running at their respective constant rate, machine X takes 2 days long [#permalink] New post 23 Jan 2015, 10:24
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LADodgers wrote:
Running at their respective constant rate, machine X takes 2 days longer to produce w widgets than machines Y. AT these rates, if the two machines together produce 5w/4 widgets in 3 days, how many days would it take machine X alone to produce 2w widgets.

A. 4
B. 6
C. 8
D. 10
E. 12

OPEN DISCUSSION OF THIS QUESTION IS HERE: running-at-their-respective-constant-rate-machine-x-takes-98599.html


For work problems one of the most important thin to know is \(rate*time=job \ done\).

Let the time needed for machine X to produce \(w\) widgets be \(t\) days, so the rate of X would be \(rate=\frac{job \ done}{time}=\frac{w}{t}\);

As "machine X takes 2 days longer to produce \(w\) widgets than machines Y" then time needed for machine Y to produce \(w\) widgets would be \(t-2\) days, so the rate of Y would be \(rate=\frac{job \ done}{time}=\frac{w}{t-2}\);

Combined rate of machines X and Y in 1 day would be \(\frac{w}{t}+\frac{w}{t-2}\) (remember we can sum the rates). In 3 days two machines together produce 5w/4 widgets so: \(3(\frac{w}{t}+\frac{w}{t-2})=\frac{5w}{4}\) --> \(\frac{w}{t}+\frac{w}{t-2}=\frac{5w}{12}\).

\(\frac{w}{t}+\frac{w}{t-2}=\frac{5w}{12}\) --> reduce by \(w\) --> \(\frac{1}{t}+\frac{1}{t-2}=\frac{5}{12}\).

At this point we can either solve quadratic equation: \(5t^2-34t+24=0\) --> \((t-6)(5t-4)=0\) --> \(t=6\) or \(t=\frac{4}{5}\) (which is not a valid solution as in this case \(t-2=-\frac{6}{5}\), the time needed for machine Y to ptoduce \(w\) widgets will be negatrive value and it's not possible). So \(t=6\) days is needed for machine X to produce \(w\) widgets, hence time needed for machine X to produce \(2w\) widgets will be \(2t=12\) days.

OR try to substitute the values from the answer choices. Remember as we are asked to find the time needed for machine X alone to produce \(2w\) widgets then the answer should be \(2t\) among answer choices: E work - \(2t=12\) --> \(t=6\) --> \(\frac{1}{6}+\frac{1}{6-2}=\frac{5}{12}\).

Answer: E.

OPEN DISCUSSION OF THIS QUESTION IS HERE: running-at-their-respective-constant-rate-machine-x-takes-98599.html
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Re: Running at their respective constant rate, machine X takes 2 days long   [#permalink] 23 Jan 2015, 10:24
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