Julio brought $42.50 to the art supply store. He bought a brush, a sketchbook, and a paint set. The brush was 1/4
 as much as the sketchbook, and the sketchbook cost 2/3
 the cost of the paint set. Julio had $4.00 left over after buying these items.


Let the cost of the paint set be denoted by $ P $. The problem provides relationships between the cost of the brush, sketchbook, and paint set:

- The cost of the sketchbook is $ \frac{2}{3}P $.
- The cost of the brush is $ \frac{1}{4} $ of the sketchbook, which gives $ \frac{1}{4} \times \frac{2}{3}P = \frac{1}{6}P $.

The total amount Julio spent can be expressed as the sum of the costs of the brush, sketchbook, and paint set:
\[
\text{Total spent} = \frac{1}{6}P + \frac{2}{3}P + P
\]

Julio had $4.00 left after his purchases, so the total amount spent is $ 42.50 - 4.00 = 38.50 $.

Thus, we have the equation:
\[
\frac{1}{6}P + \frac{2}{3}P + P = 38.50
\]

### Solving the equation:
1. Convert all terms to have a common denominator:
   \[
   \frac{1}{6}P + \frac{2}{3}P + P = \frac{1}{6}P + \frac{4}{6}P + \frac{6}{6}P = \frac{11}{6}P
   \]
2. Now solve for $ P $:
   \[
   \frac{11}{6}P = 38.50
   \]
   Multiply both sides by 6:
   \[
   11P = 231
   \]
   Divide by 11:
   \[
   P = 21
   \]

So, the cost of the paint set is $21. 

Now we can find the costs of the other items:
- The sketchbook costs $ \frac{2}{3} \times 21 = 14 $ dollars.
- The brush costs $ \frac{1}{6} \times 21 = 3.50 $ dollars.

### Summary:
- Paint set: $21
- Sketchbook: $14
- Brush: $3.50
- Julio spent $38.50, leaving him with $4.00.

Would you like further details or clarifications?

Here are 5 related questions for practice:
1. What if the brush was 1/3 the cost of the sketchbook?
2. How would the result change if the paint set was twice as expensive?
3. Can you calculate Julio's total expenditure if the paint set was discounted by 10%?
4. What would be the total if Julio bought two sketchbooks instead of one?
5. How much money would Julio need if the brush price increased by 20%?

**Tip:** When dealing with proportions, always express relationships between quantities in terms of a single variable for easier calculations.

Julio brought $42.50 to the art supply store. He bought a brush, a sketchbook, and a paint set. The brush was 1/4 as much as the sketchbook, and the sketchbook cost 2/3 the cost of the paint set. Julio had $4.00 left over after buying these items.

Learn how to solve Julio's art supply problem using algebra and proportions. This problem involves calculating the costs of a brush, sketchbook, and paint set based on given relationships. Suitable for Grades 6-8, this step-by-step solution will guide you through solving linear equations.

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