Question: This table shows a proportional relationship between the number of cups of blue paint and yellow paint used for green paint.
Cups of
Blue Paint
$(x)$
Cups
Yellow Paint
$(y)$
13
15
5 1
5
1 13
30
8 3
5
a.) Enter the number of cups of blue
paint used for 1 cup of yellow paint.
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 blue paint $(x)$ and cups of yellow paint $(y)$ used to make green paint. We want to find:
a) The number of cups of blue paint used for 1 cup of yellow paint.
b) The constant of proportionality relating $x$ and $y$.
c) An equation relating $x$ and $y$.
2. **Understand proportional relationships:** If $x$ and $y$ are proportional, then there exists a constant $k$ such that:
$$y = kx$$
or equivalently,
$$k = \frac{y}{x}$$
3. **Find the constant of proportionality $k$:** Using the first pair $(x=13, y=15)$,
$$k = \frac{y}{x} = \frac{15}{13}$$
4. **Find the number of cups of blue paint used for 1 cup of yellow paint:** Since $k = \frac{y}{x}$, we want to find $x$ when $y=1$:
$$1 = kx \implies x = \frac{1}{k} = \frac{1}{\frac{15}{13}} = \frac{13}{15}$$
So, for 1 cup of yellow paint, we use $\frac{13}{15}$ cups of blue paint.
5. **Write the equation relating $x$ and $y$:** Using $k = \frac{15}{13}$,
$$y = \frac{15}{13} x$$
6. **Summary:**
a) Number of cups of blue paint for 1 cup yellow paint is $\frac{13}{15}$.
b) Constant of proportionality $k = \frac{15}{13}$.
c) Equation relating $x$ and $y$ is:
$$y = \frac{15}{13} x$$