Widget production

Running at their respective constant rate, 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?

1. 4
2. 6
3. 8
4. 10
5. 12

Let’s collect the information in a table. I scaled the production by 4 to get rid of the fraction. The variable d is the number of days it takes machine X to produce w widgets (not 2w).

machine(s)	widgets	days
X&Y	5w	12
X	1w	d
Y	1w	d−2

To compare the productivities, we let everything work the same time T=12d(d−2), a multiple of the three times in the table.

machine(s)	widgets	days
X&Y	5d(d−2)⋅w	T
X	12(d−2)⋅w	T
Y	12d⋅w	T

We can infer that 12d+12(d−2)=5d(d−2) or 5d2−34d+24=0.

From the answer choices we conclude that d must be one of the values 2,3,4,5, or 6. Plugging in odd numbers won’t work (why?), and d≠2 because otherwise d−2=0. We only have to check 4 and 6, which reveals the correct answer choice (E).