Equivalent Ratios and Simplification — Grade 8 Unit Overview

Where this unit fits

This is a Grade 8 Number unit in the Canadian curriculum, inside Ratios, Rates and Proportions and specifically within Ratio Concepts and Comparison. It narrows the broad idea of ratio comparison to one central question: when do two ratios describe the same relationship?

For the broader parent unit, see [rea.m08.number.ratios-rates-proportions] and [rea.m08.number.ratios-rates-proportions.ratio-concepts.overview]. For a closely related overview of proportions and missing values, see [rea.m08.number.ratios-rates-proportions.equivalent-ratios.overview] and [rea.m08.number.ratios-rates-proportions.equivalent-ratios.lesson].

What you will be able to do after this unit

By the end of this Grade 8 unit, you should be able to:

• generate equivalent ratios by multiplying both terms by the same non-zero number;
• generate equivalent ratios by dividing both terms by the same common factor;
• decide whether two ratios are equivalent using scaling, ratio tables, diagrams, or division-based reasoning;
• simplify a ratio to an equivalent ratio in lowest terms;
• explain why simplifying a ratio does not change the relationship being compared;
• keep the order of terms and the units consistent when comparing ratios;
• use equivalent ratios to solve simple comparison and missing-value problems.

Prerequisite skills and earlier units

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

• Meaning and Representation of Ratios: reading ratio language, writing ratios in forms such as a:b, a to b, and a/b, and identifying what each term compares.
• Ratio Concepts and Comparison — Grade 8 Unit Overview: understanding that ratios compare quantities and that valid comparisons must keep the same meanings, order, and units.
• earlier number fluency from prior grades: multiplication facts, division facts, factors, multiples, and common factors.

If those ideas are shaky, review ratio meaning first. Equivalent-ratio work fails quickly when the learner mixes up the order of the terms or treats ratios as additive instead of multiplicative.

Core concepts

1. Equivalent ratios describe the same comparison

If 2:3 becomes 4:6, the numbers changed, but the relationship did not. Both say that for every 2 of one quantity, there are 3 of the other.

Intuition: imagine a recipe using 2 cups of juice concentrate and 3 cups of water. Doubling the recipe gives 4 cups and 6 cups. The batch is larger, but the taste stays the same because both quantities were scaled by the same factor.

2. The key idea is multiplicative scaling

Equivalent ratios come from multiplying or dividing both parts by the same non-zero number.

• 3:5 -> 6:10 by multiplying both terms by 2
• 12:20 -> 3:5 by dividing both terms by 4

This is the central habit of mind in the unit: ask, "Can I scale one ratio into the other with one common factor?"

3. Simplifying a ratio is scaling down, not changing it

To simplify 18:24, divide both terms by a common factor. Dividing by 6 gives 3:4.

This is not a new ratio with a new meaning. It is the same comparison written more simply. Lowest terms are useful because they make patterns easier to see and comparisons easier to justify.

4. Equivalent does not mean equal term-by-term

A learner may notice that 6 - 4 = 2 and 9 - 6 = 3 and try to compare by differences. That fails. Ratios are about multiplicative comparison, not additive change.

For example:

• 2:3 and 4:5 are not equivalent, even though both increase by 2 from first term to second in one direction or look close in size.
• 2:3 and 4:6 are equivalent because both terms were multiplied by 2.

5. Order matters

2:5 and 5:2 are not the same ratio. One could mean "2 red for every 5 blue," while the other means "5 red for every 2 blue." Equivalent ratios must keep the same order and meaning.

6. Units and context matter

A ratio only makes sense when you know what each part refers to.

• 3 apples : 4 oranges can be compared with 6 apples : 8 oranges.
• It cannot be directly compared with 6 oranges : 8 apples unless the order is rewritten to match.

Intuitive models that help

Ratio table

A ratio table makes scaling visible.

Juice	Water
2	3
4	6
6	9
8	12

Each row shows the same relationship.

Grouping picture

Think of 2:3 as one group of 2 matched with one group of 3. Two groups give 4:6. Three groups give 6:9. The number of groups changes; the internal comparison does not.

Simplification as undoing groups

If 12:18 is made of 6 groups of 2:3, then simplifying means finding the size of one basic group.

Step-by-step procedures

To generate an equivalent ratio

1. Start with a ratio, such as 5:7.
2. Choose a non-zero scale factor.
3. Multiply both terms by that factor.
4. Write the new ratio and keep the same order.

Example: 5:7, multiply by 3 -> 15:21

To check whether two ratios are equivalent

1. Keep the order of terms the same.
2. Ask whether both terms were multiplied or divided by the same number.
3. If yes, the ratios are equivalent.
4. If not, they are not equivalent.

Example: Are 8:12 and 2:3 equivalent?

• 8 ÷ 4 = 2
• 12 ÷ 4 = 3

Yes, they are equivalent.

To simplify a ratio

1. Find a common factor of both terms.
2. Divide both terms by that factor.
3. Repeat until no common factor greater than 1 remains.

Example: Simplify 20:28.

• Both are divisible by 2: 20:28 = 10:14
• Both are divisible by 2 again: 10:14 = 5:7
• 5 and 7 have no common factor greater than 1

So the simplified ratio is 5:7.

Suggested order of study

1. Review ratio meaning, notation, and order.
2. Build the idea of scaling a ratio up with concrete contexts such as recipes, maps, or paint mixtures.
3. Use ratio tables and simple diagrams to see families of equivalent ratios.
4. Practise checking equivalence by identifying a common scale factor.
5. Learn to simplify ratios by dividing by common factors.
6. Move to lowest terms and justify why the simplified ratio is still equivalent.
7. Apply the idea to short word problems and missing values.
8. End with mixed practice: generate, test, simplify, explain.

Fully worked examples

Example 1: Generate equivalent ratios

Write three ratios equivalent to 3:4.

Multiply both terms by the same number.

• by 2: 3:4 = 6:8
• by 3: 3:4 = 9:12
• by 5: 3:4 = 15:20

Answer: 6:8, 9:12, and 15:20

Example 2: Decide whether two ratios are equivalent

Are 10:15 and 14:21 equivalent?

Simplify each ratio.

• 10:15, divide both terms by 5 -> 2:3
• 14:21, divide both terms by 7 -> 2:3

Since both simplify to 2:3, they are equivalent.

Answer: yes, they are equivalent.

Example 3: Simplify a ratio in context

A class has 18 students wearing hats and 24 not wearing hats. Write the ratio of hats to no hats in simplest form.

Start with 18:24.

• divide both by 6 -> 3:4

So the simplified ratio of hats to no hats is 3:4.

Interpretation: for every 3 students wearing hats, there are 4 not wearing hats.

Example 4: Missing value with equivalent ratios

Complete the ratio: 4:7 = 12:x

Ask how 4 became 12.

• 4 x 3 = 12

So multiply the second term by 3 as well.

• 7 x 3 = 21

Therefore x = 21.

Common misconceptions and repairs

• Misconception: "Equivalent ratios can be found by adding the same number to both terms." Repair: Use a recipe example. Adding changes the taste; multiplying keeps the taste the same. Compare 2:3 with 3:4 and notice they are not equivalent.

• Misconception: "Simplifying changes the ratio." Repair: Show several rows in a ratio table, such as 2:3, 4:6, 6:9. Simplifying moves to a smaller row in the same family.

• Misconception: "2:5 and 5:2 are equivalent because they use the same numbers." Repair: Attach labels. 2 red : 5 blue is not the same as 5 red : 2 blue.

• Misconception: "If two ratios have the same difference, they are equivalent." Repair: Compare 3:5 and 4:6. Both differ by 2, but 3:5 does not scale to 4:6 with one common factor.

Short practice with answers

1. Write two ratios equivalent to 4:9. Answer: examples include 8:18 and 12:27.

2. Are 6:10 and 9:15 equivalent? Answer: yes. Both simplify to 3:5.

3. Simplify 14:35. Answer: divide both terms by 7 -> 2:5.

4. Complete: 5:8 = 15:x. Answer: x = 24 because the scale factor is 3.

5. Explain why 12:16 and 3:4 are equivalent. Answer: dividing both terms of 12:16 by 4 gives 3:4, so they describe the same comparison.

How to know you are ready to move on

You are ready for the next unit when you can:

• see quickly whether a ratio has been scaled correctly;
• simplify most whole-number ratios without guessing;
• explain equivalence using words such as "same multiplicative relationship" or "same scale factor";
• solve simple missing-value ratio problems without confusing additive and multiplicative thinking.

That readiness supports later Grade 8 work with proportions, unit rates, and proportional reasoning across contexts.