Comparing Ratios — Grade 8 Unit Overview

Where This Unit Fits

This is a Grade 8 Number unit in Ratios, Rates and Proportions, inside Ratio Concepts and Comparison. It focuses on a central question: when two situations are described by ratios, how can you tell which ratio shows the larger relationship, or whether they describe the same relationship?

This unit should be studied alongside the broader overviews:

• rea.m08.number.ratios-rates-proportions.ratio-concepts.overview
• rea.m08.number.ratios-rates-proportions.equivalent-ratios.overview
• rea.m08.number.ratios-rates-proportions

What You Will Be Able to Do After This Unit

By the end of this unit, you should be able to:

• decide which of two ratios is greater, or whether they are equal;
• compare ratios by scaling to equivalent ratios;
• compare ratios by rewriting them in common terms, such as the same first quantity or the same second quantity;
• explain ratio comparisons in words, tables, and simple diagrams, not only with calculations;
• tell when a comparison is meaningful and when the ratios describe different kinds of relationships;
• justify answers in everyday contexts such as recipes, team records, mixtures, maps, class surveys, and sports statistics.

Prerequisite Skills

Before this unit, you should already be comfortable with ideas from these earlier units:

• Ratio Concepts and Comparison — Grade 8 Unit Overview: rea.m08.number.ratios-rates-proportions.ratio-concepts.overview
• Equivalent Ratios and Proportions — Grade 8 Unit Overview: rea.m08.number.ratios-rates-proportions.equivalent-ratios.overview

In particular, you should already know how to:

• read and write ratios such as 3:5, 3 to 5, and 3/5 in context;
• distinguish part-to-part from part-to-whole comparisons;
• scale a ratio up or down without changing its meaning;
• recognize when two ratios are equivalent.

Core Idea

A ratio compares quantities by describing a relationship, not just two separate numbers. Comparing ratios means comparing those relationships.

For example, 2 red : 3 blue and 4 red : 6 blue look different as pairs of numbers, but they describe the same relationship because both can be read as “for every 2 red, there are 3 blue.” By contrast, 2:3 and 3:4 are not equal, because the second ratio has more of the first quantity for each amount of the second.

The main mathematical habit in this unit is: do not compare ratio numbers one by one; compare the relationship they represent.

Core Concepts

1. Equal Ratios Describe the Same Relationship

Two ratios are equal if one can be scaled to the other by multiplying or dividing both terms by the same non-zero number.

Example:

• 3:4 and 6:8 are equal because both terms were multiplied by 2.
• 5:8 and 10:16 are equal for the same reason.

Intuition: if every quantity in a situation is doubled, the balance stays the same.

2. Comparing by Common Terms

A powerful way to compare ratios is to rewrite them so one part matches.

Example: Compare 2:5 and 3:5.

• The second terms are already the same: both compare something to 5.
• Since 3 > 2, 3:5 is greater than 2:5 if the first quantity is the part being compared.

Example: Compare 3:4 and 6:10.

• Rewrite 3:4 as 6:8.
• Now compare 6:8 with 6:10.
• With the same first term, having only 8 in the second term means a larger first-to-second relationship than 6:10.

Intuition: make the situations easier to line up before judging them.

3. Comparing by Scaling to Equivalent Ratios

If common terms are not already visible, create them.

Example: Compare 4:7 and 6:9.

• Scale 4:7 to 12:21.
• Scale 6:9 to 14:21.
• Now the second terms match, so compare 12:21 and 14:21.
• Since 14:21 has more of the first quantity for the same second quantity, 6:9 is greater.

This method is especially useful when ratios do not simplify to obvious common terms right away.

4. Context Decides What “Greater” Means

Ratios must be interpreted in context.

Example:

• Mixture A has 2 cups juice : 5 cups water.
• Mixture B has 3 cups juice : 7 cups water.

If you are asking, “Which mixture is more juicy?” you compare the amount of juice for the same amount of water, or compare juice as a fraction of the whole. The larger ratio means the stronger juice flavour.

Intuition: a ratio is never just arithmetic; it answers a real comparison question.

5. Some Comparisons Are Not Meaningful Until the Ratio Type Matches

You should only compare ratios that describe the same kind of relationship.

Example:

• 3 boys : 5 girls
• 3 apples : 5 baskets

These have the same numbers, but not the same meaning. Even within one topic, you must be careful not to compare a part-to-part ratio with a part-to-whole ratio as if they were the same thing.

Common Misconceptions and Repairs

Misconception: “The bigger numbers make the bigger ratio.”

Repair: Compare relationships, not raw size. 2:3 is greater than 4:7 even though 4 and 7 are larger numbers.

Misconception: “If one term is bigger, the whole ratio is bigger.”

Repair: A ratio has two linked terms. You must consider both at once.

Misconception: “Equivalent ratios are different because the numbers changed.”

Repair: Scaling both terms by the same factor preserves the relationship.

Misconception: “Any two ratios can be compared directly.”

Repair: First check that the ratios describe the same kind of comparison and the same order of quantities.

Suggested Order of Study

A good self-study sequence is:

1. Review what a ratio means and how order matters.
2. Review equivalent ratios and scaling.
3. Compare ratios that already have a common first or second term.
4. Compare ratios by creating common terms through scaling.
5. Compare ratios in context: mixtures, class data, maps, and game statistics.
6. Practice explaining why one ratio is greater, not only which one is greater.
7. Finish with mixed review and then the mastery quiz: rea.m08.number.ratios-rates-proportions.ratio-concepts.unit-mastery-quiz

How This Connects Forward

This unit prepares you for later Grade 8 work with:

• unit rates, where one term is deliberately made 1 for easier comparison;
• proportions, where you test whether two ratios are equal;
• percent reasoning, where comparing parts to wholes becomes especially important.

If you can compare ratios confidently, then later topics like best buys, speed, density, and scale drawings will make much more sense.