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 54w widgets in 3 days, how many days would it take machine x alone to produce 2w widgets?

12 days.

Step-by-step explanation:

Let x be the time taken by machine Y to produce w widget,

So, the efficiency of machine Y = ,

Since, X takes 2 days longer to produce w widgets than Machine Y,

So, the efficiency of machine X = ,

If total work = ,

Then the time taken by machines X and Y when they work together

According to the question,

Number of days can not be negative,

Hence, the time taken by machine X to produce w widgets = x + 2 = 4 + 2 = 6 days,

Therefore the time taken by machine X to produced 2w widgets would be  12 days.