# Lesson 17

Two Related Quantities, Part 1

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

### 17.1: 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

### 17.2: Painting the Set

Lin needs to mix a specific shade of orange paint for the set of the school play. The color 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.

cups of red paint $$(r)$$ cups of yellow paint $$(y)$$ total cups of paint $$(t)$$
2 3
6
20
18
14
16
50
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.

A fruit stand sells apples, peaches, and tomatoes. Today, they sold 4 apples for every 5 peaches. They sold 2 peaches for every 3 tomatoes. They sold 132 pieces of fruit in total. How many of each fruit did they sell?

### 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.

We can see from the table that $$r$$ is always $$\frac53$$ as large as $$g$$ and that $$g$$ is always $$\frac35$$ as large as $$r$$.

green apples ($$g$$) red apples ($$r$$)
3 5
6 10
9 15
12 20

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:

$$\displaystyle r=\frac53g$$

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:

$$\displaystyle g=\frac35r$$

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 Entries

• dependent variable

The dependent variable is the result of a calculation.

For example, a boat travels at a constant speed of 25 miles per hour. The equation $$d=25t$$ describes the relationship between the boat's distance and time. The dependent variable is the distance traveled, because $$d$$ is the result of multiplying 25 by $$t$$.

For example, a boat travels at a constant speed of 25 miles per hour. The equation $$d=25t$$ describes the relationship between the boat's distance and time. The independent variable is time, because $$t$$ is multiplied by 25 to get $$d$$.