# Lesson 16: Two Related Quantities, Part 1

Let’s use equations and graphs to describe relationships with ratios.

## 16.1: Which One Would You Choose?

Which one would you choose? Be prepared to explain your reasoning.

• A 5-pound jug of honey for \$15.35 • Three 1.5-pound jars of honey for \$13.05

## 16.2: Painting the Set

Lin needs to mix a specific color of paint for the set of the school play. The color is a shade of orange that uses 3 parts yellow for every 2 parts red.

1. Complete the table to show different combinations of red and yellow paint that will make the shade of orange Lin needs.

row 1 cups of red paint $(r)$ cups of yellow paint $(y)$ total cups of paint $(t)$
row 2 2 3
row 3 6
row 4     20
row 5   18
row 6 14
row 7 16
row 8     50
row 9   42
2. Lin notices that the number of cups of red paint is always $\frac25$ of the total number of cups. She writes the equation $r=\frac25 t$ to describe the relationship. Which is the independent variable? Which is the dependent variable? Explain how you know.

3. Write an equation that describes the relationship between $r$ and $y$ where $y$ is the independent variable.

4. Write an equation that describes the relationship between $y$ and $r$ where $r$ is the independent variable.

5. Use the points in the table to create two graphs that show the relationship between $r$ and $y$. Match each relationship to one of the equations you wrote.

GeoGebra Applet P7pydzZB

GeoGebra Applet DvdeB7ZP

## Summary

Equations are very useful for describing sets of equivalent ratios. Here is an example.

A pie recipe calls for 3 green apples for every 5 red apples. We can create a table to show some equivalent ratios.

green apples (g) red apples (r)
row 1 3 5
row 2 6 10
row 3 9 15
row 4 12 20

We can see from the table that $r$ is always $\frac53$ as large as $g$ and that $g$ is always $\frac35$ as large as $r$. We can write equations to describe the relationship between $g$ and $r$.

• When we know the number of green apples and want to find the number of red apples, we can write: $$r=\frac53g$$ In this equation, if $g$ changes, $r$ is affected by the change, so we refer to $g$ as the independent variable and $r$ as the dependent variable.

We can use this equation with any value of $g$ to find $r$. If 270 green apples are used, then $\frac53 \boldcdot (270)$ or 450 red apples are used.

• When we know the number of red apples and want to find the number of green apples, we can write: $$g=\frac35r$$ In this equation, if $r$ changes, $g$ is affected by the change, so we refer to $r$ as the independent variable and $g$ as the dependent variable.

We can use this equation with any value of $r$ to find $g$. If 275 red apples are used, then $\frac35 \boldcdot (275)$ or 165 green apples are used.

We can also graph the two equations we wrote to get a visual picture of the relationship between the two quantities.

## Glossary

dependent variable

#### dependent variable

A variable representing the output of a function.

independent variable

#### independent variable

A variable representing the input of a function.