Jones can paint a car in 8 hours. Smith can paint the same car in 6 hours. They start to paint the car together. After 2 hours, Jones leaves for lunch and Smith finishes painting the car alone. How long does it take Smith to finish?

If Jones can paint a car in 8 hours, then he gets 1/8 of the job done in 1 hour. If Smith can paint the car in 6 hours, then he gets 1/6 of the job done in 1 hour. We need to find out how long it would take them to get the whole job done if they work together. Even though we are asked to find out how long it will take Smith to finish when Jones ditches him, we have to start here first. The work equation for that problem is

[tex]\frac{1}{8}+\frac{1}{6}=\frac{1}{x}[/tex]. The LCD is 24x, so mutliplying everything by 24x gives us the new equation

3x + 4x = 24 and

7x = 24 so

x = 3.428571429 hours. Working together they get 100% of the job done in 3.428571429 hours. Therefore, if 100% of the job is done in 3.428571429 hours, we can use that to find the percentage they get done working together for 1 hour:

[tex]\frac{percent}{hours}: \frac{100}{3.428571429}=\frac{x}{1}[/tex] Cross multiply to get

100% = (3.428571429)x% and divide to get

[tex]x=29\frac{1}{6}[/tex]% of the job gets done in 1 hour when they work together. But they work 2 hours together, so that means that in 2 hours, right when Jones leaves, [tex]58\frac{1}{3}[/tex] % of the car is done, leaving Smith with [tex]41\frac{2}{3}[/tex] % to finish on his own.

We go back to the beginning where we determined that Smith can finish 1/6 of the car alone in 1 hour, so he gets [tex]16\frac{2}{3}[/tex] % done per hour on his own.

If he can finish 16.66666% of the car in an hour on his own, we can use ratios to find out how long it will take him to finish the remaining 41.6666% on his own.

[tex]\frac{percent}{hours}:\frac{16.6666}{1}=\frac{41.6666}{x}[/tex] Cross multiply to get

16.6666x = 41.6666 so

x = 2.5

It will take Smith 2.5 hours to finish the job after Jones leaves.