We can use the "add up the
rates to get the combined
rate" method to build an equation. (rsarashi, I think this is what you are referring to -- in your example, 1/k is a rate; something can do 1 job in k hours, or 1/k jobs per hour)
X's Rate + Y's rate = Combined rate
Since Rate = (Work) / (Time), we get
X's Rate + Y's rate = Combined rate --->
w/(x) + w/(y) = (5w/4)/3 [x represents X's time to make w widgets, y represents Y's]
w/(x) + w/(x-2) = (5w/4)/3 [If X takes two more days, then Y takes two fewer]
1/(x) + 1/(x-2) = 5/12
Algebra looks unpleasant (though it would certainly be possible), so let's switch to working with the answer choices. Somehow we want a 12 in the denominator, so I'll start with an answer choice that seems likely to lead to that.
Let's pretend B is the answer. The question is asking how long it would take x to make 2w widgets. If X takes 6 days to make 2w, it would take 3 days to make w (that's what our x represents in our equation above, so lets plug in:
1/(3) + 1/(3-2) = 5/12 --> No good. the left is much bigger than the right. To make the left smaller, let's pick a number that will lead to bigger denominators on the left.
Let's pretend E is the answer (it will lead to smaller values on the left and seems likely to lead to a 12 in the denominator). The question is asking how long it would take X to make 2w widgets. If X takes 12 days to make 2w, it would take 6 days to make w (that's what our x represents in our equation above, so let's plug in:
1/(6) + 1/(6-2) = 5/12
1/6 + 1/4 = 5/12
2/12 + 3/12 = 5/12 --> Good. E is our answer.
More on the method used above here:
https://www.youtube.com/watch?v=GAOel-n ... TbTHvt40S0
