For example, if a painter takes 2 hours to paint a wall, this means that this painter works at a rate of

1 wall/job 2 hours

=

1 2

job per hour

If another painter takes 5 hours to paint a wall, his rate of work is ¹/₅ job per hour. When these two painters work together, their combined rate of work is ¹/₂ + ¹/₅ = ⁷/₁₀ job per hour.

To paint a wall, how long will it take the two painters working together?

	Work (job)	Time (hours)	Rate (job/hour)
Painter 1	1	2	1/2
Painter 2	1	5	1/5
Together	1	t	1/t

We form the equation and solve for t.

1/2 + 1/5	=	1/t
7/10	=	1/t
t	=	10/7

Therefore, it takes the two painters ¹⁰/₇ hours to paint a wall together.

Example 1: Man-hours

Solve.
If 4 workers can build a barn in 6 hours, how long would it have taken for 3 workers? hours

We will use x to represent our unknown.
Let x = number of hours for 3 workers to build a barn.

(4 workers)(6 hours)	=	(3 workers)x
24 worker-hours	=	3x workers
8 hours	=	x

Example 2: Pipes

Solve. Simplify all fraction answers and use simple form. For example, if the answer is 1 2/4, use 3/2.
Decimal form is also acceptable.
One pipe can fill a tank 1.5 times faster than another pipe can drain the tank. Starting with an empty tank, if both pipes are turned on, it takes 4.5 hours to fill the tank. How long does it take the slower pipe working alone to drain a full tank? hours

Since the filling pipe is 1.5 faster than the draining pipe, the draining pipe takes 1.5 times more time to drain a tank than it takes the filling pipe to fill the tank.

	Work (tank)	Time (hours)	Rate (tank/hour)
Pipe (fill)	1	f	1/f
Pipe (drain)	1	1.5f	1/1.5f
Together	1	4.5	1/4.5

Write out an equation and solve for f.

1/f - 1/1.5f	=	1/4.5
1/f - 2/3f	=	2/9	Convert decimals to fractions for ease of calculation.
9 - 6	=	2f	Multiply through by 9f
3/2	=	f	Solve for f

The faster pipe alone takes f = 3/2 or 1.5 hours to fill an empty tank. This means that the slower pipe alone takes 1.5f = 9/4 or 2.25 hours to drain a full tank.
The answer can be either hours or hours