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Running at their respective constant rates, Machine X takes

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Senior Manager
Joined: 06 Dec 2016
Posts: 251
Re: Running at their respective constant rates, Machine X takes [#permalink]

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22 Apr 2017, 06:25
Engr2012 wrote:
jasonfodor wrote:
can someone pelase explain how they got from (1/t) + (1/((t-2)) = 5/12
TO
5t^2-34t+24=0

$$\frac{1}{t} + \frac{1}{t-2} = \frac{5}{12}$$

$$\frac{2t-2}{t*(t-2)} =\frac{5}{12}$$

Cross multiplying and rearranging terms, you get ,

$$5t^2-34t+24= 0$$

Hope it helps.

Note, if you do it this way below, you are going to get the same answer.

1/x + 1/x+2 = 5/12
2x+2/x^2 + 2x = 5/12
12(2x + 2) = 5(x^2 + 2x)
24x + 24 = 5x^2 + 10x
5x^2 - 14x - 24 = 0
(5x + 6)(x-4)

x + 2
4 + 2
6

Director
Joined: 17 Dec 2012
Posts: 635
Location: India
Re: Running at their respective constant rates, Machine X takes [#permalink]

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05 Jun 2017, 17:23
Bunuel wrote:

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

1. Do all the calculations for 2w widgets
2. It takes x 4 days more to produce 2w widgets'
3. Together they take 4.8 days to produce 2w widgets
4. 1/y + 1/x=1/4.8 or 1/y + 1/(y+4) =5/24, y=8 . Therefore y+4=12.
X takes 12 days to produce 2w widgets
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Re: Running at their respective constant rates, Machine X takes [#permalink]

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08 Jun 2017, 17:06
1
Bunuel wrote:

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

We can let x = the number of days it takes Machine X to produce w widgets and y = the number of days it takes Machine Y to produce w widgets. Thus, x = y + 2 or y = x - 2. Furthermore, the rate of Machine X is w/x and the rate of Machine Y is 1/y = w/(x - 2). We are given that they can produce (5/4)w widgets in 3 days. Thus, we have:

3(w/x) + 3[w/(x - 2)] = (5/4)w

3w/x + 3w/(x - 2) = (5/4)w

Multiplying both sides by 4x(x - 2), we have:

12w(x - 2) + 12wx = 5wx(x - 2)

12(x - 2) + 12x = 5x(x - 2)

12x - 24 + 12x = 5x^2 - 10x

5x^2 - 34x + 24 = 0

(5x - 4)(x - 6) = 0

x = ⅘ or x = 6

However, x can’t be ⅘; if it were, y would be negative. Thus, x must be 6. Since it takes Machine X 6 days to produce w widgets, it will take 12 days to produce 2w widgets.

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Manager
Joined: 21 Jun 2017
Posts: 78
Running at their respective constant rates, Machine X takes [#permalink]

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11 Oct 2017, 13:37
Set the number of widgets to a preferred number. I choose 20.

(1/y + 2) + (1/y) + 1/3 = 5/4(20) = 25
2/6+y + 3/6+y + 6/6+y = 25
11/25 = 6 + y
11/25 - 6/1 = y
5/25 = 1/5 = y = machine y takes 5 days to make 25, therefore it takes 4 days to make 20
machine x takes two days more than y to produce 20 widgets; therefore, it takes machine x 2 + 4 = 6 days for 20 widgets

6 days for 20 widgets, 12 days for 40 widgets or 2w
Director
Joined: 09 Mar 2016
Posts: 610
Re: Running at their respective constant rates, Machine X takes [#permalink]

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09 May 2018, 14:56
PareshGmat 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 'X-2' 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.....

$$5x^2 - 14x - 24 = 0$$

$$5x^2 - 20x + 6x - 24 = 0$$

$$5x(x-4) + 6(x-4) = 0$$

(x-4)(5x+6) = 0

x = 4 (Ignore the -ve equation)

x+2 = 6

pushpitkc
here cant fugure out what numbers are multiplied to get -24 and add up to -14. i wanna do factorization as i did in this post, https://gmatclub.com/forum/how-many-rea ... l#p1999687 but simply these figures dont come to my mind and i dont want to use lengthy way through formula b^2 -4ac

is it some kind of new way of factorization ?

$$5x^2 - 14x - 24 = 0$$

$$5x^2 - 20x + 6x - 24 = 0$$ how to understand this ?

$$5x(x-4) + 6(x-4) = 0$$ and this

(x-4)(5x+6) = 0

how we get x = 4 ?

see ya tomorrow thanks!
BSchool Forum Moderator
Joined: 26 Feb 2016
Posts: 2837
Location: India
GPA: 3.12
Running at their respective constant rates, Machine X takes [#permalink]

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10 May 2018, 01:16
1
dave13 wrote:
PareshGmat 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 'X-2' 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.....

$$5x^2 - 14x - 24 = 0$$

$$5x^2 - 20x + 6x - 24 = 0$$

$$5x(x-4) + 6(x-4) = 0$$

(x-4)(5x+6) = 0

x = 4 (Ignore the -ve equation)

x+2 = 6

pushpitkc
here cant fugure out what numbers are multiplied to get -24 and add up to -14. i wanna do factorization as i did in this post, https://gmatclub.com/forum/how-many-rea ... l#p1999687 but simply these figures dont come to my mind and i dont want to use lengthy way through formula b^2 -4ac

is it some kind of new way of factorization ?

$$5x^2 - 14x - 24 = 0$$

$$5x^2 - 20x + 6x - 24 = 0$$ how to understand this ?

$$5x(x-4) + 6(x-4) = 0$$ and this

(x-4)(5x+6) = 0

how we get x = 4 ?

see ya tomorrow thanks!

Hi dave13

Unfortunately, there will be times when you will have to use the lengthy method

Here is a simple workaround which can be used for this problem.
We need two numbers whose sum is -14 and product is -24*5 = -120

Prime-factorizing 120, we get $$2^3 * 3 * 5$$ and there are 4*4 or 16 factors possible.
Listing down the factors, we will have 1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120. From these
factors we need to find a combination which gives a sum of -14, which is possible when x=-20 and 6

However, for bigger numbers where is it not possible to list down the factors, it will be difficult
to solve a problem using this method. I would recommend that you solve some problems using
this formula just in case you can't use the above-given method.

Hope this helps you!
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Director
Joined: 09 Mar 2016
Posts: 610
Running at their respective constant rates, Machine X takes [#permalink]

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10 May 2018, 09:13
hey pushpitkc

will you join on my bumpy ride to understand a few questions regarding my soluton in the post above

In my solution above I get the roots 4 and -1.2 are these roots correct ? whereas you get x=-20 and 6 through factorization..why ?

Also when you say "We need two numbers whose sum is -14 and product is -24*5 = -120"

$$5y^2 -14y-24=0$$ and considering that $$5y^2$$ is A $$-14y$$ is B and $$-24$$ is C

so i can conclude you mulptiply C by A ---> -24 *5 to find the numbers that add up -14 ---> can you please remind the formula.

And if equation were like this $$y^2 -14y-24=0$$ (without 5) so i simply would multiply -24 by 1 ? right ?

many thanks and have great weekend

p.s looks like I accidently deleted the solution I posted yesterday...
BSchool Forum Moderator
Joined: 26 Feb 2016
Posts: 2837
Location: India
GPA: 3.12
Re: Running at their respective constant rates, Machine X takes [#permalink]

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11 May 2018, 10:59
1
dave13 wrote:
hey pushpitkc

will you join on my bumpy ride to understand a few questions regarding my soluton in the post above

In my solution above I get the roots 4 and -1.2 are these roots correct ? whereas you get x=-20 and 6 through factorization..why ?

Also when you say "We need two numbers whose sum is -14 and product is -24*5 = -120"

$$5y^2 -14y-24=0$$ and considering that $$5y^2$$ is A $$-14y$$ is B and $$-24$$ is C

so i can conclude you mulptiply C by A ---> -24 *5 to find the numbers that add up -14 ---> can you please remind the formula.

And if equation were like this $$y^2 -14y-24=0$$ (without 5) so i simply would multiply -24 by 1 ? right ?

many thanks and have great weekend

p.s looks like I accidently deleted the solution I posted yesterday...

Hey dave13

The roots(in the previous solution) are x = 4 and $$5x = -6$$ -> $$x = -\frac{6}{5} = -1.2$$
You have solved the problem correctly by using the formula and arrived at the right solution.

Have a great weekend!
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You've got what it takes, but it will take everything you've got

Re: Running at their respective constant rates, Machine X takes   [#permalink] 11 May 2018, 10:59

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