5. Ratio & Proportion

Lesson

Proportional relationships can be represented in a table, in an equation, verbally, or graphically.

Before we discuss graphing proportional relationships, we will review the coordinate plane:

- A number plane is created by two perpendicular lines that we call an $x$
`x`-axis and a $y$`y`-axis. - The
**$x$**is the horizontal line and the`x`-axis**$y$**is the vertical line.`y`-axis - Where the two axes cross each other is labeled the origin. It has a zero value on both axes.
- Both axes have positive and negative values which are divided by the origin as shown in the diagram below.

We can create a grid from the two number lines. When labeling points on the grid, we always use the $x$`x`-value first.

We know that a ratio compares the relationship between two values; it compares how much there is of one thing compared to another. We can therefore plot pairs of ratios on a number plane.

Plotting ratios on the coordinate plane

- The first number in a ratio becomes the $x$
`x`-value on the number plane. - The second number in that ratio becomes the $y$
`y`-value on a number plane.

Plot the ratio $3:19$3:19 on a number plane.

**Think:** The first number in the ratio is $3$3 so that is the $x$`x` coordinate . The second number in the ratio is $19$19, so that will be the $y$`y` value.

**Do:** Plot the coordinate $\left(3,19\right)$(3,19).

The yellow lines represent the distances from the axes and the green dot is our solution.

Proportional relationships can also be written as linear equations.

Writing a proportion as an equation

Proportional relationships can be written generally in the form:

$y=kx$`y`=`k``x`

where $k$`k` is the constant of proportionality

We can solve proportional relationships like equations to make judgments about the relationship between two variables. We can also graph these equations.

The diagram below shows a graph where Variable $2$2 is directly proportional to Variable $1$1:

Every proportional equation has a graph similar to this one. It is a straight line, and goes through the point $\left(0,0\right)$(0,0). This is because if $x=0$`x`=0 then $y$`y` must be equal to $0$0 as well. The constant of proportionality is what determines how steep the line is.

Joe serves $2$2 hot dogs every $4$4 minutes.

Using $y$`y` for the number of hot dogs and $x$`x` for the amount of minutes that have passed, write an equation that represents this proportional relationship.

**Think:** The equation for a proportional relationship is $y=kx$`y`=`k``x`. We can substitute two of these variables to solve for the constant of proportionality, $k$`k`.

**Do:**

$y$y |
$=$= | $kx$kx |
General equation of a proportional relationship. |

$2$2 | $=$= | $k\times4$k×4 |
Start by substituting the values from the problem. |

$\frac{2}{4}$24 | $=$= | $k$k |
Solve for $k$ |

$\frac{1}{2}$12 | $=$= | $k$k |
Simplify. The constant of proportionality is $\frac{1}{2}$12. |

Now, we substitute the constant of proportionality back into the general equation:

$y$y |
$=$= | $kx$kx |
General equation of a proportional relationship. |

$y$y |
$=$= | $\frac{1}{2}x$12x |
Substitute $x$ |

**Reflect: **How long would it take Joe to serve $12$12 hot dogs?

The price for playing arcade games is shown in the graph with price vs number of games.

What does the point $\left(3,5\right)$(3,5) represent on the graph?

**Think:** Which variable does each axis represent?

**Do:**

The horizontal axis tells us the price. Since the $x$`x`-coordinate corresponds to the horizontal axis, this means the price is $\$3$$3.

Similarly, the vertical axis tells us the number of games played. The $y$`y`-coordinate corresponds to the vertical axis, so this means $5$5 games were played.

Thus, $5$5 games cost $\$3$$3.

**Reflect:** How many games could you play for $\$12$$12?

Plot $1:3$1:3 on the coordinate plane below.

- Loading Graph...

Consider the following graph:

Loading Graph...

Which of the following could be being represented by this graph and ratio?

For every $1$1 green sweet in a mix, there are $2$2 red sweets.

AFor every $2$2 green sweets in a mix, there is $1$1 red sweet.

BFor every $1$1 green sweet in a mix, there are $2$2 red sweets.

AFor every $2$2 green sweets in a mix, there is $1$1 red sweet.

B- What ratio has been plotted?
$2:1$2:1

A$1:2$1:2

B$2:1$2:1

A$1:2$1:2

B

Frank serves $8$8 cups of coffee every $9$9 minutes.

Using $y$

`y`for the number of cups of coffee, and $x$`x`for the amount of minutes that have passed, write an equation that represents this proportional relationship.

Recall that a rate is a special type of ratio that is used to compare different types of quantities.

A unit rate describes how many units of the first type of quantity corresponds to one unit of the second type of quantity.

Some common unit rates are distance per hour, cost per item, earnings per week, etc. Do you see how in each example the first quantity is related to *one* unit of the second quantity?

When we are looking at unit rates on graphs, we want to know how much the dependent ($y$`y`) variable will increase by when the independent ($x$`x`) variable is increased by one.

Consider this table:

$x$x |
$y$y |
---|---|

$1$1 | $6$6 |

$2$2 | $12$12 |

$3$3 | $18$18 |

$4$4 | $24$24 |

$5$5 | $30$30 |

Plot the line.

Loading Graph...State the unit rate of the graph you have plotted.

Consider this table:

$x$x |
$y$y |
---|---|

$2$2 | $5$5 |

$3$3 | $7.5$7.5 |

Plot the line.

Loading Graph...State the unit rate of the graph you have plotted.

Recognize and represent proportional relationships between quantities.

(d) Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.