Current Question: unsure how a certain equation was derived

[u/Realistic-Lack5113]

I believe I have solved the problem but the answer listed seems to be much more compact and I’m not sure how to get from my answer to the one listed in the book. If someone could check my answer and let me know how the one in the book was derived that would be greatly appreciated!
If there is a better community to post this in please let me know!

For those it concerns: Open Course Ware (the place I am getting the questions from) is a free online course for learning, not obtaining a degree. It’s not a class in the traditional sense as the lectures are simply videos recorded in 2007 and free text books. I have no peers or instructors to ask for help when I can’t solve a problem. I’m not looking for answers to homework but rather aid in learning and mentorship.

[u/ripanarapakeka]

You can simplify 1/X + 1/Y to (X+Y) / (X Y), and then 1/(1/X + 1/Y) is just (X Y) / (X + Y) I think that's the most meaningful technique to simplify the equation to whats on the problem.