Let's solve this problem step-by-step.

1. Define variables:

   • Let $P$ represent the cost of the paint set in dollars.
   • Then, the sketchbook costs $\frac{2}{3} \times P$.
   • The brush costs $\frac{1}{4} \times \left(\frac{2}{3} \times P\right) = \frac{1}{6} \times P$.

2. Set up the equation:

   • Joshua has $40.00 and was left with $1.50 after his purchases.
   • Therefore, the total spent on the brush, sketchbook, and paint set is: $40.00 - 1.50 = 38.50$
   • So, the equation for the total cost is: $P + \frac{2}{3}P + \frac{1}{6}P = 38.50$

3. Combine terms:

   • Convert the terms to have a common denominator: $P + \frac{2}{3}P + \frac{1}{6}P = \frac{6}{6}P + \frac{4}{6}P + \frac{1}{6}P = \frac{11}{6}P$
   • So the equation becomes: $\frac{11}{6}P = 38.50$

4. Solve for $P$:

   • Multiply both sides by $\frac{6}{11}$: $P = 38.50 \times \frac{6}{11}$ $P = 21$
   • So, the paint set costs $21.00.

5. Find the cost of the sketchbook and brush:

   • Sketchbook: $\frac{2}{3} \times 21 = 14$
   • Brush: $\frac{1}{4} \times 14 = 3.50$

6. Solution summary:

   • The brush costs $3.50.
   • The sketchbook costs $14.00.
   • The paint set costs $21.00.

If you want further clarification on any steps or have additional questions, let me know!

Here are some related questions:

1. What if Joshua had brought a different amount of money?
2. How would the answer change if the sketchbook was $\frac{3}{4}$ the cost of the paint set?
3. What would happen if Joshua bought another item with the same ratio structure?
4. How would you solve this if given only the total cost without the leftover amount?
5. Can you find a way to verify these kinds of problems without variables?

Tip: Always define each variable clearly and write down any assumptions to stay organized when solving multi-step problems!