A **proportional relationship** shows the relationship between two quantities; it compares how much there is of one thing compared to another. In other words, it is a collection of equivalent ratios.

We can use this relationship to help us write equations to solve problems. A **proportion** is an equality between 2 ratios, or an equation that states that two ratios are equal.

\displaystyle \dfrac{a}{b} = \dfrac{c}{d}

\bm{a,\,c}

represent same part or type

\bm{b,\,d}

represent same type, sum of parts, or whole

When writing a proportion we need to make sure that both numerators represent the same thing and both denominators represent the same thing.

Not only can we write proportions from equivalent ratios, but we can also solve these equations to help answer questions and discover missing values.

Consider this ratio table that shows how far Jenny can run in miles per minute:

time (min.) | 0 | 9 | 18 | 27 | 36 | 63 |
---|---|---|---|---|---|---|

distance (mi.) | 0 | 1 | 2 | 3 | 4 | â¬š |

We can solve for the missing distance by using equivalent ratios:

\displaystyle \dfrac{1}{9} | \displaystyle = | \displaystyle \dfrac{d}{63} | Let d represent the missing value |

Looking at the denominators, we must multiply 9 by 7 to get 63. To create an equivalent ratio, we need to multiply both the numerator and denominator by 7:

\displaystyle \dfrac{1 \cdot 7}{9 \cdot 7} | \displaystyle = | \displaystyle \dfrac{7}{63} | Creating an equivalent ratio |

Therefore, the missing value in the table is 7, and Jenny can run 7 miles in 63 minutes.

A more efficient algebraic method is the **Means Extremes Property**. For the proportion a:b=c:d, the **extremes** are a and d while the **means** are b and c.

We can use the Means Extremes Property to express proportions in multiple ways:

We can solve the proportion in our previous example using the Means Extremes property as well.

\displaystyle \dfrac{1}{9} | \displaystyle = | \displaystyle \dfrac{d}{63} | Equivalent ratios |

\displaystyle 1 \cdot 63 | \displaystyle = | \displaystyle 9 \cdot d | Applying the Means Extremes Property |

\displaystyle 63 | \displaystyle = | \displaystyle 9d | Simplifying the multiplication |

\displaystyle 7 | \displaystyle = | \displaystyle d | Divide each side by 9 |

This confirms that Jenny can run 7 miles in 63 minutes.

Write proportions that could be used to solve each problem.

a

Joey wants to buy 8 watermelon. He knows 3 watermelon cost \$5. Write a proportion that we could solve to find Joey's total price.

Worked Solution

b

Martin wants the ratio of black tiles to white tiles in their bathroom to be 5:7. They need 1200 total tiles. Write a proportion to figure out how many black tiles they need.

Worked Solution

Consider the proportion:

\dfrac{7}{70} = \dfrac{a}{5}

a

Create a ratio table with at least 4 entries to find the missing value in the proportion.

Worked Solution

b

Determine what x-value in this proportional relationship would have a y-value of 85.

Worked Solution

Frank serves 2 cups of coffee every 5 minutes. Write and solve a proportion for the number of cups of coffee he serves in 1 hour.

Worked Solution

Idea summary

Proportions are equivalent ratios and can generally be written in the form:\dfrac{a}{b} = \dfrac{c}{d} We can use the Means Extremes Property to represent and solve proportions: