The time it takes to paint a room in hours, t, varies inversely with the number of people working on it, n. The more people who paint the room, the less time is needed to finish it. If 3 people paint the room, it will take 8 hours to finish the work

Answer 1

In this scenario, we can model the relationship using the concept of inverse variation. The relationship can be expressed mathematically as:

\[ t = \frac{k}{n} \]

where:

• \( t \) is the time (in hours) it takes to paint the room,
• \( n \) is the number of people working on it,
• \( k \) is a constant.

Given that 3 people can paint the room in 8 hours, we can use this information to find the value of \( k \):

\[ 8 = \frac{k}{3} \]

To solve for \( k \), multiply both sides by 3:

\[ k = 8 \times 3 = 24 \]

Now we have the constant \( k = 24\).

We can now rewrite the relationship as:

\[ t = \frac{24}{n} \]

This equation allows us to determine the time taken to paint the room based on the number of people working on it.

Examples:

1. If 4 people work on it: \[ t = \frac{24}{4} = 6 \text{ hours} \]

2. If 6 people work on it: \[ t = \frac{24}{6} = 4 \text{ hours} \]

3. If 12 people work on it: \[ t = \frac{24}{12} = 2 \text{ hours} \]

Feel free to ask if you need further information or examples!