Answer
3 hours
Work Step by Step
Let the amount of time Karen takes to paint the house be x hours.
The amount of time it takes for both of them to paint the house is then $\frac{2x}{3}$ hours.
Betty can paint $\frac{1}{6}$ of the house in 1 hour, Karen can paint $\frac{1}{x}$ of the house in 1 hour, and they can paint $\frac{3}{2x}$ of the house in 1 hour together.
From here, we can set up an equation:
$\frac{1}{6} + \frac{1}{x} = \frac{3}{2x}$
$\frac{x}{6x} + \frac{6}{6x} = \frac{3}{2x}$
$\frac{x + 6}{6x} = \frac{3}{2x}$
$2x^{2} + 12x = 18x$
$2x^{2} - 6x = 0$
$2x(x -3) = 0$
$x = 0, x = 3$
Since x cannot be 0, x is equal to 3, and the amount of time it takes for Karen to paint the house is 3 hours.
