You want to paint a piece of pottery at an art studio.  The total cost is the cost of the piece plus an hourly studio fee.  There are two studios to choose from.  Write and solve an equation to determine after how many hours of painting are the total costs the same at both studios



Let's define the variables and set up the problem:

- Let $ x $ represent the number of hours of painting.
- Let $ C_1 $ be the total cost at Studio 1.
- Let $ C_2 $ be the total cost at Studio 2.
- Let $ p_1 $ be the cost of the piece at Studio 1.
- Let $ p_2 $ be the cost of the piece at Studio 2.
- Let $ h_1 $ be the hourly studio fee at Studio 1.
- Let $ h_2 $ be the hourly studio fee at Studio 2.

The total cost for each studio can be expressed as:

\[
C_1 = p_1 + h_1 \cdot x
\]
\[
C_2 = p_2 + h_2 \cdot x
\]

To find out after how many hours the total costs are the same, set $ C_1 $ equal to $ C_2 $:

\[
p_1 + h_1 \cdot x = p_2 + h_2 \cdot x
\]

To solve for $ x $, follow these steps:

1. Subtract $ p_2 $ from both sides:

\[
p_1 - p_2 + h_1 \cdot x = h_2 \cdot x
\]

2. Subtract $ h_1 \cdot x $ from both sides:

\[
p_1 - p_2 = (h_2 - h_1) \cdot x
\]

3. Divide both sides by $ (h_2 - h_1) $:

\[
x = \frac{p_1 - p_2}{h_2 - h_1}
\]

This equation gives you the number of hours $ x $ after which the total costs at both studios will be the same.

### Example
If Studio 1 charges $20 for the piece and $10 per hour, and Studio 2 charges $15 for the piece and $12 per hour, then:

- $ p_1 = 20 $
- $ p_2 = 15 $
- $ h_1 = 10 $
- $ h_2 = 12 $

Substitute into the equation:

\[
x = \frac{20 - 15}{12 - 10} = \frac{5}{2} = 2.5
\]

So, after 2.5 hours of painting, the total cost at both studios will be the same.

Would you like more details on this solution, or do you have any questions?

### 5 Related Questions:
1. How does the cost change if the hourly fee at one studio is reduced?
2. What happens if the initial cost of the piece at both studios is the same?
3. How would the equation change if one studio had a different pricing structure, such as a flat fee?
4. How does the total cost vary with the number of hours if the hourly fees are non-linear (e.g., increasing rates)?
5. What are some strategies to minimize total costs based on time spent painting?

### Tip:
Always check the pricing structure thoroughly when comparing costs, especially when they involve hourly rates, to make the most cost-effective choice!

You want to paint a piece of pottery at an art studio. The total cost is the cost of the piece plus an hourly studio fee. There are two studios to choose from. Write and solve an equation to determine after how many hours of painting are the total costs the same at both studios.

Learn how to determine after how many hours of painting the total costs at two art studios will be the same. This problem involves setting up and solving linear equations, focusing on cost analysis. Suitable for students in Grades 9-11.

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