Rates and all that ...

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In the movie Little Big League, Billy Heywood needs to solve a homework question. Bill can paint a house in five hours,and Mary can paint a house in three hours. How many hours did it take both to paint the house? Suppose there were other painters as well?

This is a rate problem!


Answer. We need to find a common basis for comparing the painters. We know that Bill paints a house in five hours and Mary paints a house in three hours. This means that

We need the rates equation:

For our problem, we want just one house painted. So,


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littlebigleague__18.png Now its your turn. Use the rates equation above.

  1. Bob paints a house in 4 hours and Ted paints a house in 3 hours. How long does it take them to paint two houses? (Hint. The wrong answer: $3+4=7 $ hours. Use the rate equation!)

  2. Sean paints a house in 6 hours, Gary paints a house in 2 hours, and Susan paints a house in 3 hours. Working together, how long does it take them to paint a house?

  3. Mary and Sally paint a house in $6$ hours working together and Mary paints a house by herself in $2$ hours. How long does it take Sally to paint a house by herself?

  4. (From the Greek Anthology, c. 500 CE) Of the four spouts, one fills the whole tank in one day, one in two days, one in three days, and one in four days. What time will it take all four to fill the tank?

  5. The case of the messy mathematician. A physicist and a mathematician can clean a house in 6 hours; an engineer and the mathematician in 3 hours; and the physicist and the engineer in 1 hour and 12 minutes. How long would it take the physicist alone? (from Frank Morgan's Chat page.) What can we say about the mathematician?



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