Question 201997
 Let x=amount of days it takes Y to produce w widgets
Then Y produces at the rate of w/x widgets per day
x+2=amount of days it takes X to produce w widgets
Then X produces at the rate of w/(x+2) widgets per day
Together the two machines work at the rate of w/x + w/(x+2)=(w(x+2)+wx)/x(x+2) widgets per day
But we are told that the two machines together work at the rate of (5/12)w widgets per day ((5/4)w widgets in 3 days =(5/12)w widgets per day))

Now our first equation to solve is:
(w(x+2)+wx)/x(x+2)=(5/12)w multiply each side by x(x+2)
w(x+2)+wx=(5/12)wx(x+2) simplify
wx+2w+wx=(5/12)wx^2+(10/12)wx
2wx+2w=(5/12)wx^2+(10/12)wx multiply each term by 12
24wx+24w=5wx^2+10wx subtract 24wx and 24w from each side
5wx^2-14wx-24w=0  Assuming w is not equal to zero, divide each term by w
5x^2-14x-24=0  quadratic in standard form
If we use the quadratic formula to solve, we get:
x=(14+-26)/10 or
x=40/10=4---number of days it takes Y to produce w widgets
and
x=-14/10----Disregard.  Days in this problem are positive
Now for X
x+2=amount of days it takes X to produce w widgets or
x+2=4+2=6----number of days it takes X to produce w widgets
So X produces 2w widgets in 6*2=12 days--------------ans
Another way to look at it: X produces at the rate of w/(x+2) widgets per day or
w/(4+2)=w/6 widgets per day so in 12 days X produces (w/6)*12=2w widgets

Hope this helps---ptaylor