Problem Solving

It takes machine A x hours to manufacture a deck of cards that machine B can manufacture in 1/x hours. If machine A operates alone for y hours and is then joined by machine B until 100 decks are finished, for how long will the two machines operate simultaneously?

100y – x
--------------
x2 + 1


100x – y
--------------
x2 + 1


100y – x3 – x
------------------
x2 + 1


100y – x2y – y
---------------------
x2 + 1


100x
------------
x2 + 1

In 1 hour A finishes 1/x of the deck and B finishes X decks.

When A and B work together in 1 hour they could finish (1/x + x) of the deck or (x^2 + 1)/x.

It takes x/(x^2+1) hrs to complete one deck.

After "A" has worked alone for "y" hours, remaining number of decks would be (100-y/x), i,e (100x-y)/x.

1 decks is completed in x/(x^2+1) hrs, hence (100x-y)/x will be completed in
(100x-y)*x/(x^2+1)*x = 100x-y/(x^2+1) hours.