Question

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?

Answer 1

Answer:

The expression that shows how long the two machines will operate simultaneously is:

$\frac{y(100x-z)}{x+y}$

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:

$\frac{z}{x}$

So, the remaining decks are given by:

$100-\frac{z}{x}=\frac{100x-z}{x}$

Then, the combined rate of machines A and B would be:

$\frac{1}{x} +\frac{1}{y} =\frac{x+y}{xy}$

The work-rate formula is:

$Amount= Rate \times Time$

Hence, the time that the two machines work simultaneously is:

$Time=\frac{Amount}{Rate}$

$Time=\frac{Amount}{Rate} =\frac{\frac{100x-z}{x} }{\frac{x+y}{xy} } ={\frac{100x-z}{x} \times \frac{xy}{x+y}=\frac{y(100x-z)}{x+y}$