Question

Machines X and Y work at their respective constant rates. How many more hours does it take machine Y, working alone, to fill a production order of a certain size than it takes machine X, working alone?

1. Machine X and Y, working together, fill a production order of this size in two thirds the time that Machine X does.2. Machine Y, working alone, fills a production order of this size in twice the time that Machine X, working alone, does.

Answer 1

1)
`\Delta t=-0.5*t_(X)`

2)
`\Delta t=2*t_(X)`

Explanation:

1) X=production rate of X (units/hour)

Y=production rate of Y (units/hour)

C=order (units)

t=working time (hour)

`(X+Y)*t_(toghether)=C`

`X*t_(X)=C`

`t_(toghether)=2/3*t_(X)`

Combining:

`(X+Y)*2/3*t_(X)=X*t_(X)`

`(X+Y)*2/3=X`

`Y*2/3=X-2/3*X`

`Y=2X`

In time:
`t_(Y)=0.5*t_(X)`

Difference of time:
`\Delta t=0.5*t_(X)-t_(X)=-0.5*t_(X)`

2)
`Y*t_(Y)=C`

`X*t_(X)=C`

`t_(Y)=2*t_(X)`

Difference of time:
`\Delta t=2*t_(X)-0.5*t_(X)=2*t_(X)`