The number of hours it takes to paint a house varies inversely with the number of painters working on the job. Suppose 7 painters can paint a house in 15 hours.

Let x = number of painters working on the job, y = number of hours to paint the house, and k = the constant of variation.

a) Find the constant of variation.

Given that the number of hours \( y \) to paint a house varies inversely with the number of painters \( x \), the relationship between \( x \) and \( y \) can be expressed as:

\[
y = \frac{k}{x}
\]

where \( k \) is the constant of variation.

From the problem, we know that when there are 7 painters (\( x = 7 \)), it takes 15 hours to paint the house (\( y = 15 \)). Using this information, we can substitute into the equation to find \( k \).

\[
15 = \frac{k}{7}
\]

To solve for \( k \), multiply both sides by 7:

\[
k = 15 \times 7 = 105
\]

Thus, the constant of variation is \( k = 105 \).

Would you like further clarification on how this inverse variation works? Here are some related questions:

1. How would the time change if 5 painters were working on the job?
2. What is the formula for the number of hours if there are 10 painters?
3. How does increasing the number of painters affect the time to complete the job?
4. What is the time taken if the number of painters is doubled to 14?
5. If only 3 painters are available, how long will it take to paint the house?

**Tip:** Inverse variation means as one quantity increases, the other decreases proportionally.

The number of hours it takes to paint a house varies inversely with the number of painters working on the job. Suppose 7 painters can paint a house in 15 hours. a) Find the constant of variation.

Discover how to find the constant of variation in an inverse relationship where 7 painters take 15 hours to paint a house. This problem covers the concept of inverse variation and proportionality, ideal for students in Grades 8-10.

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