Math CentralQuandaries & Queries


Question from frustrated, a parent:

Working alone, Colleen can paint a house in 2 hours less than James. working together, they can paint the house in 10 hours. How long would it take James to paint the house by himself?
Collen: 1/x-2 times 10

James: 1/x times 10

10/x-2 + 10/x = 1

10x + 10x -20 = x(x-2)

0= x^2 -22x + 20 They have the answer as 21 hours. From my previous work as you see, how are they getting 21 hours?

If James needs x hours to paint the house he paints 1/x houses per hour. Colleen paints 1/(x-2) houses per hour. In 10 hours together they paint 10/x + 10/(x-2) houses, but you're told this sum = 1 house, i.e., 10/x + 10/(x-2) = 1. Cross multiply by x(x-2) to get 10(x-2) + 10x = x(x-2), or

x2 - 22x + 20 = 0 as you said.

thus, using the general quadratic, x = (1/2)(22 ± sqrt(484-80)) = (1/2)(22 ± sqrt(404)) which does not equal 21 in any way but it is close to it as sqrt404 is about 20 giving x about (1/2)(22+20) = 21. The answer is not 21 hours, but it's close.


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