Problem Solving

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 (5/4)w widgets in 3 days, how many days would it take machine X alone to produce 2w widgets?

4
6
8
10
12

This can be solved algebraically, but it gets a bit nasty.

I would use reverse engineering: work backwards to find out which of the answer choices fit all restrictions in the question.

I'd start by plugging in an easy number for w (here easy means a common factor of our other numbers). In the attached work, I made w=60 widgets and worked from there.

The right answer(E) is the answer that agrees with "Machine X takes 2 days longer to produce w widgets than Machine Y". The answer is E

Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machines Y. AT these rates, if the two machines together produce 5w/4 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

Let w=12.
In 3 days, the number of widgets that must be produced = (5/4)w = (5/4)*12 = 15 widgets.
To produce 15 widgets in 3 days, the required rate = 15/3 = 5 widgets per day.

We can plug in the answers, which represent the time for X to produce 2w=24 widgets.

Answer choice C: 8 days for X to produce 24 widgets
Here, the time for X to produce 12 widgets = 4 days.
Since X takes 2 days longer than Y, the time for Y to produce 12 widgets = 2 days.
Rate for X = w/t = 12/4 = 3 widgets per day.
Rate for Y = w/t = 12/2 = 6 widgets per day.
Combined rate for X+Y = 3+6 = 9 widgets per day.

Since the required rate = 5 widgets per day, X and Y are working at almost TWICE the required rate.
Since X and Y need to work MUCH MORE SLOWLY, X needs to take MUCH LONGER to produce 2w widgets.

Answer choice E: 12 days for X to produce 24 widgets
Here, the time for X to produce 12 widgets = 6 days.
Since X takes 2 days longer than Y, the time for Y to produce 12 widgets = 4 days.
Rate for X = w/t = 12/6 = 2 widgets per day.
Rate for Y = w/t = 12/4 = 3 widgets per day.
Combined rate for X+Y = 2+3 = 5 widgets per day.
Success!

The correct answer is E.