To set up the equation for the problem, we will first determine the rates at which each person paints the room.
- Person A can paint the room in 6 hours, so their rate is \( \frac{1}{6} \) of the room per hour.
- Person B can paint the room in 7 hours, so their rate is \( \frac{1}{7} \) of the room per hour.
When they work together, their combined rate is the sum of their individual rates. Thus, their combined rate is:
\[ \text{Combined rate} = \frac{1}{6} + \frac{1}{7} \]
To find a common denominator and combine these fractions, we can use 42 (the least common multiple of 6 and 7):
\[ \frac{1}{6} = \frac{7}{42} \quad \text{and} \quad \frac{1}{7} = \frac{6}{42} \]
So the combined rate becomes:
\[ \text{Combined rate} = \frac{7}{42} + \frac{6}{42} = \frac{13}{42} \]
This means that together, A and B can paint \( \frac{13}{42} \) of the room in one hour.
If they work together for \( t \) hours, the amount of the room they can paint is:
\[ \text{Amount of room painted} = \left(\frac{13}{42}\right) t \]
Since they will complete the room when this amount equals 1 (the whole room), we can set up the equation:
\[ \left(\frac{13}{42}\right) t = 1 \]
This is the equation that can be used to solve the problem.