A work problem is one in which a specific job is done in a certain length of time when a uniform rate of work is assumed. For example, if it takes a man 10 hours to paint a room,    then his rate of work is $\frac{1}{10}$ of the room per hour. 

To solve a work problem, we need to multiply the rate of work by the time to obtain the fractional part of work completed. Like in the example above, if the painter works for 7 hours, then the fractional part of the work completed is $\frac{7}{10}$.

Work Problem
John can paint a room in 12 hours while Steve can paint the same room in 10 hours. How long will it take to paint the room if John and Steve work together?

Solution
Since we wish to know how long it takes John and Steve to paint the room together, we make the following representation:

$\mathbf{x:}$  the number of hours in the time to paint the room when John and Steve are working together

Because the two painters complete the work together (they paint the same room), the fractional part of the work done by John plus the fractional part of the work done by Steve equals 1. We make the following table below to get expressions representing these fractional parts of the work.

Make a Table

We obtain the following equation by setting the sum of the last entries in the table equal to 1:
\[ \frac{x}{12} + \frac{x}{10} =1 \]
\[ 60 \cdot \frac{x}{12} + 60 \cdot \frac{x}{10} = 60 \cdot 1 \]
\[ 5x + 6x  = 60 \]
\[ 11x = 60 \]
\[ x = \frac{60}{11} \]

Hence it takes John and Steve $\frac{60}{11}$ hours to paint the room together.

Check
The fractional part part of the work done by John is $\frac{60}{11} \cdot \frac{1}{12} = \frac{5}{11}$ and the fractional part done by Steve is $\frac{60}{11} \cdot \frac{1}{10} = \frac{6}{11}$; and $\frac{5}{11} + \frac{6}{11} = 1$.