Question: Use the information given to complete the table. Then, write an equation in the form $y = kx$ to represent the situation.
While walking through the art museum, Josh notices that for every six sculptures, there are ten paintings.
How many paintings, $y$, are there for $x$ number of sculptures?
Enter the answer in each space provided. Use numbers instead of words.
Sculptures Paintings
[blank] 5
[blank] 30
48 [blank]
72 [blank]
108 [blank]
[blank] 250
186 [blank]
1. **State the problem:** We know that for every 6 sculptures, there are 10 paintings. We want to complete the table and find an equation relating paintings $y$ to sculptures $x$ in the form $y = kx$.
2. **Find the constant of proportionality $k$:** Since $y$ is proportional to $x$, we have $$y = kx$$
Given 6 sculptures correspond to 10 paintings, substitute to find $k$:
$$10 = k \times 6$$
Divide both sides by 6:
$$k = \frac{10}{6} = \frac{\cancel{10}}{\cancel{6}} \to \frac{5}{3}$$
So, $$k = \frac{5}{3}$$.
3. **Write the equation:**
$$y = \frac{5}{3} x$$
4. **Complete the table:**
- When paintings $y=5$, find sculptures $x$:
$$5 = \frac{5}{3} x \implies x = \frac{5}{\frac{5}{3}} = 5 \times \frac{3}{5} = 3$$
- When paintings $y=30$, find sculptures $x$:
$$30 = \frac{5}{3} x \implies x = 30 \times \frac{3}{5} = 18$$
- When sculptures $x=48$, find paintings $y$:
$$y = \frac{5}{3} \times 48 = 5 \times 16 = 80$$
- When sculptures $x=72$, find paintings $y$:
$$y = \frac{5}{3} \times 72 = 5 \times 24 = 120$$
- When sculptures $x=108$, find paintings $y$:
$$y = \frac{5}{3} \times 108 = 5 \times 36 = 180$$
- When paintings $y=250$, find sculptures $x$:
$$250 = \frac{5}{3} x \implies x = 250 \times \frac{3}{5} = 150$$
- When sculptures $x=186$, find paintings $y$:
$$y = \frac{5}{3} \times 186 = 5 \times 62 = 310$$
**Final completed table:**
| Sculptures | Paintings |
|------------|------------|
| 3 | 5 |
| 18 | 30 |
| 48 | 80 |
| 72 | 120 |
| 108 | 180 |
| 150 | 250 |
| 186 | 310 |
**Answer:** The equation relating paintings to sculptures is $$y = \frac{5}{3} x$$.