Unit 4, Lesson 16
Two Related Quantities, Part 1
Accelerated 6
Student Lesson Summary
Equations are very useful for representing the relationship in a set of equivalent ratios. Here is an example.
A cider 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 is always as large as and that is always as large as .
| green apples () | red apples () |
|---|---|
| 3 | 5 |
| 6 | 10 |
| 9 | 15 |
| 12 | 20 |
We can write equations to describe the relationship between and .
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When we know the number of green apples and want to find the number of red apples, we can write:
In this equation, if changes, is affected by the change, so we refer to as the independent variable and as the dependent variable.
We can use this equation with any value of to find . If 270 green apples are used, then or 450 red apples are used.
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When we know the number of red apples and want to find the number of green apples, we can write:
In this equation, if changes, is affected by the change, so we refer to as the independent variable and as the dependent variable.
We can use this equation with any value of to find . If 275 red apples are used, then or 165 green apples are used.
To help us see the relationship between the two quantities, we can also create two graphs, one graph that corresponds to each equation.
A graph that represents a ratio of two quantities. The graph has a horizontal axis labeled number of green apples and the numbers 1 through 15 are indicated, The vertical axis is labeled number of red apples and the numbers 1 through 20 are indicated. The following four points are indicated on the graph: 3 comma 5, 6 comma 10, 9 comma 15, and 12 comma 20.
A graph that represents a ratio of two quantities. The graph has a horizontal axis labeled number of red apples and the numbers 1 through 15 are indicated. The vertical axis is labeled number of green apples and the numbers 1 through 20 are indicated. The following three points are indicated on the graph: 5 comma 3, 10 comma 6, and 15 comma 9.
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 describes the relationship between the boat's distance and time. The dependent variable is the distance traveled because is the result of multiplying 25 by .
A graph of 10 points plotted in the coordinate plane with the origin labeled "O". The horizontal t axis is labeled "time in hours". The numbers 0 through 10, in increments of 2, are indicated, and there are vertical gridlines midway between. The vertical axis is labeled "distance traveled in miles". The numbers 0 through 250, in increments of 25, are indicated, and there are horizontal gridlines midway between. The data are as follows: 1 comma 25. 2 comma 50. 3 comma 75. 4 comma 100. 5 comma 125. 6 comma 150. 7 comma 175. 8 comma 200. 9 comma 225. 10 comma 250.