Question

problems 2–4: the table shows the ratios of paprika, onion powder, and cayenne pepper in a spice rub recipe.
paprika (mg) | onion powder (mg) | cayenne pepper (mg)
60 | 24 | 16
90 | 36 | 24

1. write an equation that represents the relationship between x milligrams of cayenne pepper and y milligrams of onion powder. show or explain your thinking.
2. graph the relationship on the coordinate plane.
3. how much onion powder is needed for 64 milligrams of cayenne pepper? show or explain your thinking.

Explanation:

Step1: Find the ratio

First, find the ratio of onion powder to cayenne pepper from the table. For the first row: onion powder is 24 mg, cayenne pepper is 16 mg. The ratio is $\frac{24}{16}=\frac{3}{2}$. Check with the second row: onion powder 36 mg, cayenne pepper 24 mg. $\frac{36}{24}=\frac{3}{2}$. So the ratio of onion powder (let's say $x$) to cayenne pepper (let's say $y$) is $\frac{x}{y}=\frac{3}{2}$, which can be rewritten as $x = \frac{3}{2}y$ or $y=\frac{2}{3}x$. Wait, no, the question is for 64 mg of cayenne pepper, find onion powder. So let's define: let $x$ be onion powder, $y$ be cayenne pepper. From the table, when $y = 16$, $x = 24$; $y = 24$, $x = 36$. So the relationship is proportional. The constant of proportionality $k=\frac{x}{y}=\frac{24}{16}=\frac{3}{2}$. So $x = k\times y=\frac{3}{2}y$.

Step2: Calculate onion powder for y=64

Now, we need to find $x$ when $y = 64$. Substitute $y = 64$ into the equation $x=\frac{3}{2}y$. So $x=\frac{3}{2}\times64$. Calculate $\frac{3}{2}\times64 = 3\times32 = 96$. Let's verify with the ratio. If $y = 16$, $x = 24$ (16×1.5=24). 64×1.5=96. So that makes sense.

Answer:

96 milligrams