another way: logical reasoning.Honestly, I still not catch up you logic flow. So I use the traditional one. Anyway, Do you mind using "back-solving" method for this?
1. "machine X takes 2 days longer to produce w widgets than machine Y" - machine X works slowly that machine Y
2. If machine Y work simultaneously, both machine would produce >4w widgets (Y faster than X)
3. So, 5w/4 in 3 days and >4w in ----> >9.6 days
4. in ~10 day Y can produce ~ 4w*2/(10-2) ~ w more widgets than X
5. therefore, answer has to be close to 3*(4w+w)/ (5w/4) = 12
It took ~30sec in my mind and is one of fast ways for solving work problems along with back-solving.
How did you get that N=4 without solving the equation?
X is the time machine X works w job
Y is the time machine Y works w job
X =Y+2 --> Y=X-2
(w/X) + (w/(X-2) = 5w/12
5X^2 -34X + 24 =0
Machine X do w job in 6 hours, so 2w, corresponding time is 12 hours