Work rate problem

[selango]

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 5/4w 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

OA E

[gmatjedi]

x= rate of work
rate of work of machine x= 1 / x+2
rate of work of machine y= 1 / x
cumulative rate by formula:
1/machine 1 + 1/machine 2 = 1/ cumulative work

(1/(x+2)) +1/x =1/r

Key is to recognize that this is to make 1 widget

Working together, x and y make 5/4 widgets in 3 days
Need to figure out how many days x and y together make 1 widget
5/4 / 3 = 1/x
x=12/5 days to make 1 widget

replace r in equation above with 12/5 and solve for x

(x+x+2)/(x(x+2))=1/(12/5)
12/5 * (2x+2)/(x^2+2x) = 1
24x+24=5x^2+10x
(5x+6)(x-4)=0
x =4
so rate of x
is 1/(4+2) = 1 widget every 6 days
therefore 2 widgets in 12 days

[sanju09]

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 5/4w 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

OA E

If Machine X alone takes n days to produce 2 w widgets, then it would take ½ n days to produce w widgets, and hence Machine Y alone would take ½ n - 2 days to produce w widgets. Working together they would take ½ n (½ n - 2) / (½ n + ½ n - 2) days to produce w widgets, or they would take ¼ n (n - 4) / (n - 2) days to produce w widgets; or they would take 5/4 × ¼ n (n - 4) / (n - 2) days to produce 5/4 w widgets.

Or 5/4 × ¼ n (n - 4) / (n - 2) = 3

Or 5 n (n - 4) = 48 (n - 2)

See through the choices, A is impossible, start from C, feel like going down, [spoiler]E works![/spoiler]

[liferocks]

I have used the options to reach the ans.
First let W=1..and the number of days x complets 2 machines is 2x..
So the question gives x and y completes 5/4=1+1/4 machines in 3 days
x completes 1 machine in x days and y in x-2 days

option 1>2x=4 or x=2..so x-2=0..this is the number of days for y..not possible ..eliminated
option 2> 2x=6..x=3..x-2=1..so in 3 days y makes 3 machines,,,eliminated
option 3>2x=8...x=4..x-2=2..so y makes in 3 days 3/2 machine>5/4 machines...eliminated
optiin 4>2x=10,x=5..x-2=3..so in 3 days x n y makes 1+1/5 machines<1+1/4...eliminated

option E is the choice.