Question

Betty and Karen have been hired to paint the houses in a new development. Working together the women can paint a house in two-thirds the time that it takes Karen working alone. Betty takes 6h to paint a house alone. How long does it take Karen to paint a house working alone?

Answer 1

In general, to find the time it takes two people, B and K to work together, use the following formula:

\(\large \frac{BK}{B + K} = t\) where

B = Betty
K = Karen
t = time it takes when Betty and Karen are working together

Answer 2

Adding B = 6 and \(t = \frac{2K}{3}\) to the formula we get:

\[\large \frac{6K}{6 + K} = \frac{2K}{3}\]

Answer 3

Now, just cross multiply and continue solving for K

Answer 4

3(6K) = 2K(6 + K)
18K = 12K + 2K^2
0 = 12K - 18K + 2K^2
0 = -6K + 2K^2
0 = 2K^2 - 6K
0 = 2K(K - 3)

2K = 0
K - 3 = 0

K = 3

Therefore, it takes Karen 3 hours to paint a house while working alone.