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 of the time it takes Karen when working alone. Betty takes 6 hours to paint a house by herself. Required: How long does it take Karen to paint a house on her own?

To find out how long it takes Karen to paint a house on her own, we'll set up an equation based on the work rates of Betty and Karen.

Let's denote the following: - The time it takes for Karen to paint a house alone is 'k' hours. - The time it takes for Betty and Karen to paint a house together is two-thirds of Karen's time, which is (2/3)k hours.

When working together, their rates add up, so we can state that the total work done by both is equal to the work done by each one alone. The work done by each person (or together) painting a house is considered 1 complete house.

Betty's rate is 1 house per 6 hours, so her work rate per hour is 1/6 house per hour.

Karen's rate is 1 house per k hours, so her work rate per hour is 1/k house per hour.

The combined rate of Betty and Karen working together is 1 house per (2/3)k hours, so their work rate per hour is 1/((2/3)k) = 3/(2k) houses per hour.

Now we can set up the work rate equation:

Betty's rate + Karen's rate = Combined rate (1/6) + (1/k) = (3/(2k))

Solving for k, we multiply every term by 6k to clear the fractions: k + 6 = (9k)/2

Now, multiply both sides by 2 to get rid of the fraction on the right: 2k + 12 = 9k

Subtract 2k from both sides: 12 = 7k

Divide both sides by 7 to solve for k: k = 12/7

Converting this to a mixed number, since we usually measure time in whole hours and parts of an hour as minutes: k = 1 5/7 hours

To convert 5/7 of an hour to minutes, multiply by 60: (5/7) * 60 minutes = 428.57 minutes, which rounds to approximately 43 minutes.

Therefore, it takes Karen about 1 hour and 43 minutes to paint a house on her own.