can you help me with math please.

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 14 h to paint a house alone. How long does it take Karen to paint a house working alone?

Answer 1

It's usually helpful to treat work rate problems in terms of work performed per unit time, not time required to complete the entire job.

Betty can paint a house in 14 hours, so she can paint 1/14 of a house in one hour.

Karen can paint a house in x hours, and the two women working together can paint the house in 2x/3 hours.  So Karen can paint 1/x of a house in one hour and both can do the job in 3/2x hours.

Add the "work per hour" figures:

1/14 + 1/x = 3/2x

Common denominator is 14x.

x/14x + 14/14x = 21/14x

Disregard the denominators.

x + 14 = 21

x = 7

Now let's see if the answer makes sense.  

We already knew Betty could paint a house in 14 hours, or 1/14 of a house in one hour.

We now know Karen can paint a house in 7 hours, or 1/7 of a house in one hour.

1/14 + 1/7 = 3/14.  The two women can paint 3/14 of a house in an hour, or an entire house in 14/3 or 4 2/3 hours.

4 2/3 is 2/3 of 7.