Question

Set up the equation which would be used to solve the following problem. Do not actually solve the equation.
Working​ alone, a person A can paint a room in 6 hr. Person B can paint the same room working alone in 7 hr. How long will it take them if they work​ together? (Let t represent the time they work​ together.)

Answer 1

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.