# combining rates - I; Joe & Sam paint a wall

- Author:
- Judah L Schwartz

- Topic:
- Algebra

Here is an app that lets you explore a problem that many students have difficulty with:
Joe can paint a wall by himself in four hours.
Sam can paint the same wall by himself in two hours
How long will it take them to paint the wall if they work together?
The problem is shown in the {time, walls} plane
Assuming the start at the same time, the app shows the fraction of the wall
painted at any time by Joe and by Sam. It can also show the fraction of the wall
painted at any time when the two of them painting together.
You can use the app to find the answer to the question very simply.
Now, write an equation relating the rates at which
Joe and Sam can each paint the wall and
the length of time it takes them to do so.
CHALLENGE –
a. How is the PURPLE function related to the GREEN and BLUE functions?
b. Use this app to pose and solve a different problem – for example, Joe and Sam can each paint the wall alone in 3 hours.
c. Use this app to pose and solve a different problem – Joe can paint the wall alone in 2 hours, and together Joe and Sam can paint the wall in one hour. How long would it take Sam to paint the wall if he were painting by himself?
d. Make up a new problem like this in which Joe paints 3 times as fast as Sam.
e. Make up and solve a problem in which Joe, Sam and Mary all work on painting the wall.

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