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

The relationship described in the problem can be represented by the formula for inverse variation, which states that:

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

where:

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

From the information given, we know that when \( n = 3 \) (3 people painting), it takes \( t = 8 \) hours to finish. We can use this information to find the constant \( k \).

Substituting the known values into the equation:

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

To find \( k \), we can multiply both sides by 3:

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

Now that we have the constant \( k \), we have the complete equation that describes the relationship between time and the number of people working:

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

This equation indicates how the time required to paint the room varies with the number of people working on it. For example, if you want to know how long it would take 4 people to paint the room, you would substitute \( n = 4 \) into the equation:

\[ t = \frac{24}{4} = 6 \text{ hours} \]

In summary, the time it takes to paint the room, \( t \), is given by:

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