Question

It takes Nate five hours to paint a wall, and it takes Jill seven hours to paint the same wall. About how many hours will it will take if Nate and Jill work together to paint the wall?

Answer 1

One very effective way to work these problems is to look at unit rates. Nate paints a wall in 5 hours, which means in 1 hour, he paints 1/5 of a wall (1 wall / 5 hours = 1/5 wall/hour). Similarly, Jill works at a rate of 1/7 wall/hour.

Together, assuming as these problems do that they can work together with no loss of efficiency, they will do
\[\frac{1}{5} + \frac{1}{7}\] walls every hour.

\[\frac{1}{5} + \frac{1}{7} = \frac{7}{7}*\frac{1}{5} + \frac{5}{5}*\frac{1}{7}\](making a common denominator)
\[\frac{7}{7}*\frac{1}{5} + \frac{5}{5}*\frac{1}{7} = \frac{7}{35} + \frac{5}{35} = \frac{7+5}{35} = \frac{12}{35}\]
Nate and Jill working together will complete \(\dfrac{12}{35}\) of a wall every hour. You'll need to figure out from that how many hours it will take to do the whole wall, if 1 hour gets you \(\dfrac{12}{35}\) of a wall.