Q:

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?

Accepted Solution

A:
Answer:The expression that shows how long the two machines will operate simultaneously is:[tex]\frac{y(100x-z)}{x+y}[/tex]Step-by-step explanation:We know that:x: hours to manufacture a deck of cards for machine Ay: hours to manufacture a deck of cards for machine Bz: hours that machine A operates aloneThe number of decks manufactured only by machine A is:[tex]\frac{z}{x}[/tex]So, the remaining decks are given by:[tex]100-\frac{z}{x}=\frac{100x-z}{x} [/tex]Then, the combined rate of machines A and B would be:[tex]\frac{1}{x} +\frac{1}{y} =\frac{x+y}{xy}[/tex]The work-rate formula is:[tex]Amount= Rate \times Time[/tex]Hence, the time that the two machines work simultaneously is:[tex]Time=\frac{Amount}{Rate}[/tex][tex]Time=\frac{Amount}{Rate} =\frac{\frac{100x-z}{x} }{\frac{x+y}{xy} } ={\frac{100x-z}{x} \times \frac{xy}{x+y}=\frac{y(100x-z)}{x+y}[/tex]