Question: This table shows a proportional relationship between the number of cups of cilantro and tomatoes used for pico de gallo salsa.
a.) Enter the number of cups of cilantro used for 1 cup of tomatoes.
b.) Enter the constant of proportionality that relates the quantities $x$ and $y$.
c.) Enter an equation that relates the quantities.
1. **State the problem:** We have a proportional relationship between cups of cilantro ($x$) and cups of tomatoes ($y$) given by the table:
| Cups of Cilantro ($x$) | Cups of Tomatoes ($y$) |
|------------------------|------------------------|
| $1 \frac{1}{5} = \frac{6}{5}$ | $3 \frac{3}{5} = \frac{18}{5}$ |
| $3 \frac{2}{5} = \frac{17}{5}$ | $10 \frac{1}{5} = \frac{51}{5}$ |
2. **Find the number of cups of cilantro used for 1 cup of tomatoes:**
Since the relationship is proportional, the ratio $\frac{x}{y}$ is constant.
Calculate $\frac{x}{y}$ for the first pair:
$$\frac{\frac{6}{5}}{\frac{18}{5}} = \frac{6}{5} \times \frac{5}{18} = \frac{6}{18} = \frac{1}{3}$$
This means for 1 cup of tomatoes, cilantro used is $\frac{1}{3}$ cups.
3. **Find the constant of proportionality:**
The constant of proportionality $k$ relates $x$ and $y$ by $x = ky$.
From above, $k = \frac{1}{3}$.
4. **Write the equation relating $x$ and $y$:**
$$x = \frac{1}{3} y$$
5. **Verify with second pair:**
Calculate $\frac{x}{y}$ for second pair:
$$\frac{\frac{17}{5}}{\frac{51}{5}} = \frac{17}{5} \times \frac{5}{51} = \frac{17}{51} = \frac{1}{3}$$
Confirms proportionality constant.
**Final answers:**
a.) Number of cups of cilantro for 1 cup of tomatoes is $\frac{1}{3}$.
b.) Constant of proportionality $k = \frac{1}{3}$.
c.) Equation relating quantities: $$x = \frac{1}{3} y$$