Rate Problem - Machine X and Y HELP

First, note that for something like this, the answer should be true regardless of the value of w. Hence, we can choose any non-negative value for w. Let's say w=1. Let's say machine Y takes y days to make w widgets. This would mean that machine X takes y+2 days to make w widgets. Substituting in 1 for w, we can say that machine Y makes 1 widget every y days, and machine X makes 1 widget every y+2 days. In work-rate-time problems like these, you can find a person or machine's rate by dividing the number of things done by the amount of time it takes them, so we can say machine Y works at a rate of 1/y widgets per day and machine X works at a rate of 1/(y+2) widgets per day. Therefore, in 3 days, machine Y makes 3/y widgets, and machine X makes 3/(y+2) widgets. Combined, we are told that they can make 5w/4 widgets in 3 days, so we can write: 3/y+3/(y+2)=5/4. Multiply both sides of the equation by the LCD, 4y(y+2) in order to cancel out the fractions: 3(4)(y+2)+3(4)(y)=5y(y+2). Distributing and simplifying, this equation becomes 5y^2-14y-24=0. This factors as (5y+6)(y-4)=0. y must be positive, so it must be the case that y=4.

So, how long does it take machine X to make 2 widgets? if y=4, it makes 1 widget in 6 days, so it can make 2 widgets in 12 days.

Ans: E

gmatpup wrote:In the GMAT quant review (2nd ed) I came across a question that I just cannot seem to figure out. Please help!!

173. 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

The answer is E but I just don't see how!

Thanks so much for any explanation

I posted an alternate approach here:

https://www.beatthegmat.com/how-many-day ... 11577.html