Question

If Abe paints a fence alone, he can complete the task in x hours. If Billy paints the same fence alone, he can complete the task in y hours. If Abe and Billy paint the fence together, they can complete the task in z hours. Complete the equation for z by entering a single fraction in terms of x and y that can be used to find the number of hours it will take Abe and Billy to paint the fence together.

Answer 1

It will take them
`(12)/(5)` hours (or 2.4 hours) to paint the fence together.

Step-by-step explanation:

The equation for painting the fence together can be expressed as:

`(1x)/(1y)=(1)/(z)`

This equation represents the combined rate of work for Abe and Billy. The fraction
`(1)/(x)` represents the portion of the fence that Abe can complete in an hour, while the fraction
`(1)/(y)` represents the portion of the fence that Billy can complete in an hour.

When they work together, their combined rate of work is represented by the fraction
`(1)/(z)`.

To find the number of hours it will take them to paint the fence together, solve the equation for z.

For example, if Abe takes 4 hours (x = 4) to paint the fence alone, and Billy takes 6 hours (y = 6) to paint the fence alone, the equation becomes:

`(1)/(4)+(1)/(6)=(1)/(z)`

Simplifying the equation gives:

`(3)/(12)+(2)/(12)=(1)/(z)`

`(5)/(12)=(1)/(z)`