rates/widget problem with sharing - please help!!

Dear clearmountain,
First of all, I love your screenname. You may find this blog article helpful:
https://magoosh.com/gmat/2012/gmat-work-rate-questions/

Here: each is assigned x widgets to make, and Dave takes y away from Mike's load, so Dave makes x + y and Mike must make x - y.

(amount) = (rate)(time) ====> A = RT

Dave's rate = amount/time = x/5

Mike's rate = x/7

For Dave: A = RT ---> x+y = (x/5)*T ---> 5(x+y)/x = T

For Mike: A = RT ---> x-y = (x/7)*T ---> 7(x-y)/x = T

Both T's are equal, so set these equations equal:

5(x+y)/x = 7(x-y)/x ---- then multiply by x --->

5(x+y) = 7(x-y) ---- then distribute --->

5x + 5y = 7x - 7y --- then combine terms --->

12y = 2x

We want to solve for "the number of widgets ... that Mike was assigned to make" = x, in terms of y. Divide by 2.

6y = x

So, the number of widgets each was assigned to make, x, is equal to 6y.

Does that make sense?

Mike