It takes machine A x hours to manufacture a deck of cards that machine B can manufacture in y hours. If machine A operates alone for z hours and is then joined by machine B until 100 decks are finished, for how long will the two machines operate simultaneously?
a) (100xy - z)/(x+y)
b) y(100x - z)/(x+y)
c) (x+y)/(100xy - z)
d) xz(100y - z)/(x+y+z)
e) (x+y-z)/(100xy)
Let x = 1 hour.
Since A takes 1 hour to produce a deck, A's rate = 1 deck per hour.
Let y = 2 hours.
Let A complete the ENTIRE JOB.
Since A produces 1 deck per hour, the time for A to produce all 100 decks = 100 hours.
Thus, z=100.
The question stem asks for the time that A and B work together.
Since A completes the entire job, the number of hours that A and B work together = 0. This is our target.
Now plug x=1, y=2 and z=100 into the answers to see which yields our target of 0.
A quick scan reveals that only
B works:
y(100x - z)/(x+y) = 2(100*1 - 100)/(1+2) = 0.
The correct answer is
B.
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