Question

Christopher can paint the interior of his house in 15 hours. if he hires cynthia to help him they can do the same job together in 9 hours. if he lets cynthia work alone, how long will it take her to paint the interior of his house

Answer 1

If Christopher can paint his whole house in 15 hours, then he can get 1/15 of the job done in a single hour. If he hires Cynthia to help and they get the job done together (the sum of their efforts) in 9 hours, they can get 1/9 of the job done in a single hour. We are looking for how long Cynthia would take to do it on her own. Remember we are adding their efforts together and setting that equal to the total, so the equation is $\frac{1}{15}+\frac{1}{x}=\frac{1}{9}$. What we are going to do here, what makes the most sense in solving for x, is to move the term with x in it to the other side of the equation and move the 1/9 over by subtraction. $\frac{1}{15}-\frac{1}{9}=-\frac{1}{x}$. Of course before we can subtract those 2 fractions we need a common denominator, which happens to be 45: $\frac{3}{45}-\frac{5}{45}=-\frac{1}{x}$ and $-\frac{2}{45}=-\frac{1}{x}$. We will cross multiply to get 2x = 45 and divide to find that it takes Cynthia 22.5 hours to do the job if she had to do it alone.