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Running at their respective constant rates, Machine X takes [#permalink]
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The Official Guide For GMAT® Quantitative Review, 2ND EditionRunning at their respective constant rates, Machine X takes 2 days longer to produce w widgets than Machine Y. At these rates, if the two machines together produce (5/4)w 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 Problem Solving Question: 173 Category: Algebra; Applied problems Page: 85 Difficulty: 600 GMAT Club is introducing a new project: The Official Guide For GMAT® Quantitative Review, 2ND Edition  Quantitative Questions ProjectEach week we'll be posting several questions from The Official Guide For GMAT® Quantitative Review, 2ND Edition and then after couple of days we'll provide Official Answer (OA) to them along with a slution. We'll be glad if you participate in development of this project: 1. Please provide your solutions to the questions; 2. Please vote for the best solutions by pressing Kudos button; 3. Please vote for the questions themselves by pressing Kudos button; 4. Please share your views on difficulty level of the questions, so that we have most precise evaluation. Thank you!
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SOLUTIONRunning at their respective constant rates, Machine X takes 2 days longer to produce w widgets than Machine Y. At these rates, if the two machines together produce (5/4)w 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 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 \(t2\) days, so the rate of Y would be \(rate=\frac{job \ done}{time}=\frac{w}{t2}\); Combined rate of machines X and Y in 1 day would be \(\frac{w}{t}+\frac{w}{t2}\) (remember we can sum the rates). In 3 days two machines together produce 5w/4 widgets so: \(3(\frac{w}{t}+\frac{w}{t2})=\frac{5w}{4}\) > \(\frac{w}{t}+\frac{w}{t2}=\frac{5w}{12}\). \(\frac{w}{t}+\frac{w}{t2}=\frac{5w}{12}\) > reduce by \(w\) > \(\frac{1}{t}+\frac{1}{t2}=\frac{5}{12}\). At this point we can either solve quadratic equation: \(5t^234t+24=0\) > \((t6)(5t4)=0\) > \(t=6\) or \(t=\frac{4}{5}\) (which is not a valid solution as in this case \(t2=\frac{6}{5}\), 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  \(2t=12\) > \(t=6\) > \(\frac{1}{6}+\frac{1}{62}=\frac{5}{12}\). Answer: E.
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Running at their respective constant rates, Machine X takes [#permalink]
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Time taken by Machine Y = t days Time taken by Machine X = (t+2) days Rate of Machine \(X = \frac{w}{t+2}\) Rate of Machine \(Y = \frac{w}{t}\) \(\frac{5w}{4}\) widgets produced in 3 days; so \(\frac{5w}{12}\) produced in 1 day Equation setup will be \(\frac{w}{t+2} + \frac{w}{t} = \frac{5w}{12}\) Solving, \(5t^2  14t  24 = 0\) t = 4 Time taken by Machine X = t+2 = 6 for w So for 2w; it will be 12
Answer = E
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Re: Running at their respective constant rates, Machine X takes [#permalink]
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23 Mar 2014, 07:13
SOLUTIONRunning at their respective constant rates, Machine X takes 2 days longer to produce w widgets than Machine Y. At these rates, if the two machines together produce (5/4)w 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 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 \(t2\) days, so the rate of Y would be \(rate=\frac{job \ done}{time}=\frac{w}{t2}\); Combined rate of machines X and Y in 1 day would be \(\frac{w}{t}+\frac{w}{t2}\) (remember we can sum the rates). In 3 days two machines together produce 5w/4 widgets so: \(3(\frac{w}{t}+\frac{w}{t2})=\frac{5w}{4}\) > \(\frac{w}{t}+\frac{w}{t2}=\frac{5w}{12}\). \(\frac{w}{t}+\frac{w}{t2}=\frac{5w}{12}\) > reduce by \(w\) > \(\frac{1}{t}+\frac{1}{t2}=\frac{5}{12}\). At this point we can either solve quadratic equation: \(5t^234t+24=0\) > \((t6)(5t4)=0\) > \(t=6\) or \(t=\frac{4}{5}\) (which is not a valid solution as in this case \(t2=\frac{6}{5}\), 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  \(2t=12\) > \(t=6\) > \(\frac{1}{6}+\frac{1}{62}=\frac{5}{12}\). Answer: E.
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Re: Running at their respective constant rates, Machine X takes [#permalink]
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16 May 2014, 06:22
QE: 5t^2  14t  24 = 0, How do we get solution = 4?



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Re: Running at their respective constant rates, Machine X takes [#permalink]
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Alternate Solution without algebra using POE. If it takes 3 days to produce 5/4 widgets, it will take 3*(4/5) = 12/5 days to produce 1 widget. If two machines were working at the same rate it would take, 2*12/5 i.e. approx 5 days days for x to produce 1 widget. Since machine X is slower it will take more than 5 days to produce 1 widget and subsequently more than 10 days to produce 2 widgets. The only answer more than 10 is 12 days. Answer E
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Re: Running at their respective constant rates, Machine X takes [#permalink]
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Back tracking method: We have to find the number of days it takes Machine X alone to produce 2w widgets. Answer options are as below:
(A) 4 (B) 6 (C) 8 (D) 10 (E) 12
We are given that X and Y produce (5/4)w in 3 days and also given that X take 2 days more than Y. So to find corresponding value of X and Y use the answer options. First half the value of X to find number of days it takes Machine X to complete *W* widgets and find the value of Y by reducing 2 from all answer options.
> X to complete W widgets (A) X=2 and Y= 22 =0 (B) X=3 and Y= 32 =1 (C) X=4 and Y= 42 =2 (D) X=5 and Y= 52 =3 (E) X=6 and Y= 62 =4
Now we have all the values of X and Y. Find out the combination X and Y, whose values will produce (5/4)w for 3 days of work.
3(1/6+1/4)=>5/4>Option (E) gives this value so correct answer is (E)>12.



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Re: Running at their respective constant rates, Machine X takes [#permalink]
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20 Oct 2014, 09:39
Hi,
I am new here and I have solved this math correctly. However, here's my problem:
If I consider the number of days required to produce 'W' widgets as 'X' and 'X2' for X and Y respectively, I end up with the equation 5t^2  34t + 24 which yields me 2 solutions, one of which is correct (x=6)
However, when I take 'X' and 'Y' as 'X+2' and 'X', which are the same as above, because either ways, X is taking 2 days longer, I end up with the equation 5x^2  14x  24 which has no solutions (I hope I'm not making some stupid mistake here)
Can someone please clarify this and correct me where I'm wrong.



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Re: Running at their respective constant rates, Machine X takes [#permalink]
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muhtasimhassan wrote: Hi,
I am new here and I have solved this math correctly. However, here's my problem:
If I consider the number of days required to produce 'W' widgets as 'X' and 'X2' for X and Y respectively, I end up with the equation 5t^2  34t + 24 which yields me 2 solutions, one of which is correct (x=6)
However, when I take 'X' and 'Y' as 'X+2' and 'X', which are the same as above, because either ways, X is taking 2 days longer, I end up with the equation 5x^2  14x  24 which has no solutions (I hope I'm not making some stupid mistake here)
Can someone please clarify this and correct me where I'm wrong. Refer my post above; I did in the same way..... As far as solving the equation, refer below..... \(5x^2  14x  24 = 0\) \(5x^2  20x + 6x  24 = 0\) \(5x(x4) + 6(x4) = 0\) (x4)(5x+6) = 0 x = 4 (Ignore the ve equation) x+2 = 6
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Re: Running at their respective constant rates, Machine X takes [#permalink]
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20 Oct 2014, 20:20
muhtasimhassan wrote: Hi,
I am new here and I have solved this math correctly. However, here's my problem:
If I consider the number of days required to produce 'W' widgets as 'X' and 'X2' for X and Y respectively, I end up with the equation 5t^2  34t + 24 which yields me 2 solutions, one of which is correct (x=6)
However, when I take 'X' and 'Y' as 'X+2' and 'X', which are the same as above, because either ways, X is taking 2 days longer, I end up with the equation 5x^2  14x  24 which has no solutions (I hope I'm not making some stupid mistake here)
Can someone please clarify this and correct me where I'm wrong. It doesn't matter whether you take the number of days as Days taken by machine X = X and Days taken by machine Y = X2 or Days taken by machine X = X+2 Days taken by machine Y = X Either way, you will get one valid value for X and you will need to manipulate that to get your answer. You need to find the number of days that will be taken by machine X to make 2w widgets. In first case, you get the equation as \(5X^2  34X + 24 = 0\) and get X = 6 as the only valid value (X = 4/5 gives X2 as negative but number of days cannot be negative). Machine X takes X days i.e. 6 days to make w widgets. So it will take 12 days to make 2w widgets. In the second case, you get \(5x^2  14x  24 = 0\) which is \(5x^2  20x + 6x  24 = 0\) 5x(x  4) + 6(x  4) = 0 (x4)(5x + 6) = 0 x = 4 This is the number of days taken by machine Y to make w widgets. So machine X takes 4+2 = 6 days to make w widgets. It will take 12 days to make 2w widgets. Either way, your answer will be the same.
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20 Oct 2014, 20:39
PareshGmatVeritasPrepKarishmaThanks a lot. Careless of me to not have noticed such a simple solution to the equation 5x^2  14x  24 = 0



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VeritasPrepKarishma wrote: muhtasimhassan wrote: Hi,
I am new here and I have solved this math correctly. However, here's my problem:
If I consider the number of days required to produce 'W' widgets as 'X' and 'X2' for X and Y respectively, I end up with the equation 5t^2  34t + 24 which yields me 2 solutions, one of which is correct (x=6)
However, when I take 'X' and 'Y' as 'X+2' and 'X', which are the same as above, because either ways, X is taking 2 days longer, I end up with the equation 5x^2  14x  24 which has no solutions (I hope I'm not making some stupid mistake here)
Can someone please clarify this and correct me where I'm wrong. It doesn't matter whether you take the number of days as Days taken by machine X = X and Days taken by machine Y = X2 or Days taken by machine X = X+2 Days taken by machine Y = X Either way, you will get one valid value for X and you will need to manipulate that to get your answer. You need to find the number of days that will be taken by machine X to make 2w widgets. In first case, you get the equation as \(5X^2  34X + 24 = 0\) and get X = 6 as the only valid value (X = 4/5 gives X2 as negative but number of days cannot be negative). Machine X takes X days i.e. 6 days to make w widgets. So it will take 12 days to make 2w widgets. In the second case, you get \(5x^2  14x  24 = 0\) which is \(5x^2  20x + 6x  24 = 0\) 5x(x  4) + 6(x  4) = 0 (x4)(5x + 6) = 0 x = 4 This is the number of days taken by machine Y to make w widgets. So machine X takes 4+2 = 6 days to make w widgets. It will take 12 days to make 2w widgets. Either way, your answer will be the same. Just wanted to point out that this problem can be solved quite easily without solving a quadratic. Our equation for x is given by: \(\frac{1}{x} + \frac{1}{x2} = \frac{5}{12}\). \(\frac{x2 + x} {x(x2)} = \frac{5}{12}\). \(\frac{2x2}{x(x2)} = \frac{5}{12}\). \(\frac{2(x1)}{x(x2)} = \frac{5}{12}\). \(\frac{(x1)}{x(x2)} = \frac{5}{24}\). We see immediately that by letting \(x=6\) we find our solution of x. Therefore, since we're looking for the value of 2x, we see that the answer must be 12. Answer: E



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Re: Running at their respective constant rates, Machine X takes [#permalink]
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26 Mar 2015, 00:59
Bunuel wrote: SOLUTION
Running at their respective constant rates, Machine X takes 2 days longer to produce w widgets than Machine Y. At these rates, if the two machines together produce (5/4)w 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
Any questions similar to this one somewhere? The regular Rate problems are quite easy, but as soon as they start mixing it up, adding variables, I'm clueless as to how to approach things.



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26 Mar 2015, 03:49
MensaNumber wrote: Alternate Solution without algebra using POE.
If it takes 3 days to produce 5/4 widgets, it will take 3*(4/5) = 12/5 days to produce 1 widget.
If two machines were working at the same rate it would take, 2*12/5 i.e. approx 5 days days for x to produce 1 widget. Since machine X is slower it will take more than 5 days to produce 1 widget and subsequently more than 10 days to produce 2 widgets. The only answer more than 10 is 12 days. Answer E +1 Kudos !! Have a doubt , why did you discount 'w' while calculating time require to produce 1 widget ? If it takes 3 days to produce 5/4 widgets, it will take 3*(4/5) = 12/5 days to produce 1 widget. shldn't it be \(\frac{12}{5w}\) ?
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Re: Running at their respective constant rates, Machine X takes [#permalink]
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10 May 2015, 03:48
Lucky2783 wrote: MensaNumber wrote: Alternate Solution without algebra using POE.
If it takes 3 days to produce 5/4 widgets, it will take 3*(4/5) = 12/5 days to produce 1 widget.
If two machines were working at the same rate it would take, 2*12/5 i.e. approx 5 days days for x to produce 1 widget. Since machine X is slower it will take more than 5 days to produce 1 widget and subsequently more than 10 days to produce 2 widgets. The only answer more than 10 is 12 days. Answer E +1 Kudos !! Have a doubt , why did you discount 'w' while calculating time require to produce 1 widget ? If it takes 3 days to produce 5/4 widgets, it will take 3*(4/5) = 12/5 days to produce 1 widget. shldn't it be \(\frac{12}{5w}\) ? Hi, Correction  If it takes 3 days to produce 5/4 widgets, it will take 1/3*(4/5) = 4/15 days to produce 1 widget. Thanks
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Re: Running at their respective constant rates, Machine X takes [#permalink]
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26 Jul 2015, 18:32
can someone pelase explain how they got from (1/t) + (1/((t2)) = 5/12 TO 5t^234t+24=0



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Re: Running at their respective constant rates, Machine X takes [#permalink]
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26 Jul 2015, 18:48
jasonfodor wrote: can someone pelase explain how they got from (1/t) + (1/((t2)) = 5/12 TO 5t^234t+24=0 \(\frac{1}{t} + \frac{1}{t2} = \frac{5}{12}\) \(\frac{2t2}{t*(t2)} =\frac{5}{12}\) Cross multiplying and rearranging terms, you get , \(5t^234t+24= 0\) Hope it helps.



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26 Jul 2015, 20:36
jasonfodor wrote: can someone pelase explain how they got from (1/t) + (1/((t2)) = 5/12 TO 5t^234t+24=0 You can also give the quadratic route a miss. Once you get (1/t) + (1/(t2)) = 5/12, you can try out some values for t. t is an integer since it is one of the given 5 options. To get 12 in denominator on the right hand side, t*(t2) should give you 12 or a multiple. 12 = 4*3 or 6*2 (not of the form t*(t2)) 24 = 6*4 (this is possible) Check 1/6 + 1/4 = 10/24 = 5/12 You get the value of t. Or you can also try substituting the value of t from each option.
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