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The problem presented asks us to find the cost of one quart of blue paint and one quart of yellow paint, given the following information:

1. **Lime green paint** is made by mixing 2 parts blue to 6 parts yellow, making 2 gallons of lime green.
   - Lime green paint sells for $29 per gallon.

2. **Ocean blue paint** is made by mixing 5 parts blue to 3 parts yellow, making 2 gallons of ocean blue.
   - Ocean blue paint sells for $32 per gallon.

The task is to determine the cost of:
- One quart of blue paint.
- One quart of yellow paint.

### Solution Setup:
Let's assume:
- $ b $ = cost per quart of blue paint.
- $ y $ = cost per quart of yellow paint.

Since there are 4 quarts in a gallon, the price per gallon of blue and yellow paint will be $ 4b $ and $ 4y $, respectively.

Now, we will form two equations based on the given ratios for lime green and ocean blue paint.

#### 1. Lime Green Paint
Lime green uses a 2:6 ratio of blue to yellow, which makes 2 gallons of paint. Therefore, in terms of quarts, the paint mixture contains:
- 2 parts blue = 2 quarts blue paint.
- 6 parts yellow = 6 quarts yellow paint.

Thus, the total cost for 1 gallon (which is half of this mixture) of lime green paint will be:
\[
\frac{2}{8}(4b) + \frac{6}{8}(4y) = 29
\]
This simplifies to:
\[
b + 3y = 29 \quad \text{(Equation 1)}
\]

#### 2. Ocean Blue Paint
Ocean blue uses a 5:3 ratio of blue to yellow, which makes 2 gallons of paint. In terms of quarts, the paint mixture contains:
- 5 parts blue = 5 quarts blue paint.
- 3 parts yellow = 3 quarts yellow paint.

The total cost for 1 gallon of ocean blue paint will be:
\[
\frac{5}{8}(4b) + \frac{3}{8}(4y) = 32
\]
Simplifying this equation gives:
\[
2.5b + 1.5y = 32 \quad \text{(Equation 2)}
\]

### Solving the System of Equations
We have the system:
1. $ b + 3y = 29 $
2. $ 2.5b + 1.5y = 32 $

Now let's solve this system of equations to find the values of $ b $ and $ y $. I will compute that next.The solution to the system of equations gives:

- The cost of one quart of blue paint $ b $ is **$8.75**.
- The cost of one quart of yellow paint $ y $ is **$6.75**.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How would the costs change if the price of lime green paint increased by 10%?
2. What is the total cost of making 5 gallons of lime green paint?
3. How would the cost per quart change if the ratio of blue to yellow was altered?
4. Can you set up a similar problem for another paint color using a different ratio?
5. How could we calculate the cost of mixing blue and yellow paint to create custom colors?

**Tip**: When solving systems of equations, always double-check your coefficients and simplify ratios for easier calculations.

A hardware store mixes paints in a ratio of two parts blue to six parts yellow to make two gallons of lime green. A ratio of five parts blue to three parts yellow makes two gallons of ocean blue. A gallon of lime green sells for $29, and a gallon of ocean blue sells for $32. Find the cost of 1 quart of blue paint and the cost of 1 quart of yellow paint.

Learn how to solve a paint mixture problem involving the cost of blue and yellow paints using ratios and systems of linear equations. This problem explores algebraic concepts and is suitable for students in Grades 9-11.

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