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 A
y: hours to manufacture a deck of cards for machine B
z: hours that machine A operates alone
The 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]
