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What are Ratios?

Representing Equivalent Ratios

Solving Ratios and Rate Problems

A ratio is an association between two or more quantities. For example, say we have a drink recipe made with cups of juice and cups of soda water. Ratios can be represented with diagrams like those below.

Here are some correct ways to describe this diagram:

The ratios 6:4, 3:2, and 12:8 are equivalent because each ratio of juice to soda water would make a drink that tastes the same.

Here is a task to try with your student:

There are 4 horses in a stall. Each horse has 4 legs, 1 tail, and 2 ears.

Complete each statement.

Solution:

There are different ways to represent ratios.

Let’s say the 6th grade class is selling raffle tickets at a price of $6 for 5 tickets. Some students may use diagrams with shapes to represent the situation. For example, here is a diagram representing 10 tickets for $12.

Drawing so many shapes becomes impractical. Double number line diagrams are easier to work with. The one below represents the price in dollars for different numbers of raffle tickets all sold at the same rate of $12 for 10 tickets.

Raffle tickets cost $6 for 5 tickets.

Over the course of this unit, your student has learned to use the language of ratios and to work with ratios using representations like diagrams and double number lines. In the final sections of the unit, they use tables to organize equivalent ratios. Double number lines are hard to use in problems with large amounts. Let’s think about an example we saw before: the 6th grade class is selling raffle tickets at a price of $6 for 5 tickets. If we tried to extend the double number line below to represent the price of 300 raffle tickets, it would take 5 times more paper!

A table is a better choice to represent this situation. Tables of equivalent ratios are useful because you can arrange the rows in any order. For example, a student may find the price for 300 raffle tickets by making the table shown.

Although students can choose any representation that helps them solve a problem, it is important that they get comfortable with tables because they are used for a variety of purposes throughout high school and college mathematics courses.

At a constant speed, a train travels 45 miles in 60 minutes. At this rate, how far does the train travel in 12 minutes? If you get stuck, consider creating a table.

9 miles. Possible strategy:

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.