Equivalent Ratios and Simplification

Place in the unit

This lesson is for Grade 8 Number: Ratios, Rates and Proportions, in the subtopic Ratio Concepts and Comparison.

Use this record as the main instructional lesson for the topic. For broader placement and sequencing, see:

• Equivalent Ratios and Simplification — Grade 8 Unit Overview
• Generating Equivalent Ratios by Scaling — Grade 8 Unit Overview
• Ratios, Rates and Proportions (Grade 8, Number)

For end-of-unit checking, use the linked mastery quiz rather than treating this lesson as an assessment.

Learning goal

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

• recognize when two ratios are equivalent,
• generate equivalent ratios by scaling up or down,
• simplify a ratio to lowest terms,
• solve for a missing term in an equivalent-ratio statement,
• explain why the procedure works using multiplicative reasoning.

Prerequisite idea

This lesson assumes you already know what a ratio means and how to read forms such as a:b, a to b, and a/b as a comparison. If that meaning is not secure, review the separate ratio-meaning record first rather than trying to learn this lesson procedurally.

Intuition: keeping the same comparison

A ratio compares two quantities.

Suppose a picture uses 2 red tiles for every 3 blue tiles. That ratio is 2:3.

If you make a larger copy of the same picture by doubling every group, you get 4 red for every 6 blue. That ratio is 4:6.

Nothing about the comparison changed. The picture is larger, but the red-to-blue balance is the same.

So 2:3 and 4:6 are equivalent ratios.

Visual description

Imagine drawing two bars:

• first bar: 2 equal red blocks beside 3 equal blue blocks,
• second bar: 4 equal red blocks beside 6 equal blue blocks.

The second bar is just a doubled version of the first. Each colour was scaled by the same factor, so the relationship stayed the same.

That is the core idea:

A ratio stays equivalent when both parts are multiplied or divided by the same non-zero number.

Formal definition and notation

Ratios a:b and c:d are equivalent if they represent the same comparison.

You can create an equivalent ratio by:

• multiplying both terms by the same non-zero factor, or
• dividing both terms by the same non-zero factor, when both terms divide evenly.

Examples:

• 3:5 = 6:10 because both terms were multiplied by 2.
• 12:18 = 2:3 because both terms were divided by 6.

Important notation

A ratio may be written as:

• a:b
• a to b
• a/b

In this lesson, these forms all describe the same comparison, but the colon form a:b is often clearest when simplifying ratios.

Why the rule works

If a ratio is a:b, then scaling both parts by the same non-zero number k gives (ak):(bk).

The new ratio compares the same-sized groups, just with k copies of each part.

Example:

• 3:4 means 3 units for every 4 units.
• multiply both by 5 -> 15:20
• this means 15 units for every 20 units, which is 5 copies of the original comparison.

The comparison did not change; only the size of the group changed.

Simplifying a ratio

To simplify a ratio means to write an equivalent ratio using smaller numbers.

A ratio is in lowest terms when the two terms have no common factor greater than 1.

Procedure: simplify a ratio

1. Write the ratio clearly in one form, usually a:b.
2. Find the greatest common factor (GCF) of both terms.
3. Divide both terms by the GCF.
4. Check that the new terms have no common factor greater than 1.

Example of the procedure only

Simplify 18:24.

1. Ratio: 18:24
2. GCF of 18 and 24 is 6.
3. Divide both terms by 6: 18 ÷ 6 : 24 ÷ 6 = 3:4
4. 3 and 4 share no factor greater than 1, so 3:4 is lowest terms.

Generating equivalent ratios

Scale up

To make a larger equivalent ratio, multiply both terms by the same number.

Example:

• 5:7
• multiply both by 3
• 15:21

Scale down

To make a smaller equivalent ratio, divide both terms by the same number.

Example:

• 20:28
• divide both by 4
• 5:7

Fully worked examples

Worked example 1: recognizing equivalence

Are 4:6 and 10:15 equivalent?

Step 1: Simplify 4:6.

• GCF of 4 and 6 is 2.
• 4:6 = 2:3

Step 2: Simplify 10:15.

• GCF of 10 and 15 is 5.
• 10:15 = 2:3

Step 3: Compare the simplified forms.

• Both simplify to 2:3.

Conclusion: 4:6 and 10:15 are equivalent ratios.

Worked example 2: generating an equivalent ratio

Write three equivalent ratios for 3:8.

We multiply both terms by the same factor.

Using factor 2:

• 3 x 2 : 8 x 2 = 6:16

Using factor 3:

• 3 x 3 : 8 x 3 = 9:24

Using factor 5:

• 3 x 5 : 8 x 5 = 15:40

Conclusion: Examples of equivalent ratios are 6:16, 9:24, and 15:40.

Worked example 3: simplifying to lowest terms

Simplify 36:54.

Step 1: Find the GCF of 36 and 54.

• Factors of 36 include 1, 2, 3, 4, 6, 9, 12, 18, 36
• Factors of 54 include 1, 2, 3, 6, 9, 18, 27, 54
• Greatest common factor is 18.

Step 2: Divide both terms by 18.

• 36 ÷ 18 = 2
• 54 ÷ 18 = 3

So 36:54 = 2:3.

Step 3: Check lowest terms.

• 2 and 3 have no common factor greater than 1.

Conclusion: The simplified ratio is 2:3.

Worked example 4: solving for a missing term

Find x in 7:9 = x:27.

We need the second term 9 to become 27.

Step 1: Determine the scale factor.

• 27 ÷ 9 = 3

So the ratio was multiplied by 3.

Step 2: Apply the same factor to the first term.

• 7 x 3 = 21

So x = 21.

Check:

• 7:9 = 21:27
• both are the same comparison

Conclusion: x = 21.

Worked example 5: simplifying in context

A drink mix uses 12 scoops of powder for 18 cups of water. Write the ratio of powder to water in simplest form.

Step 1: Write the ratio.

• powder : water = 12:18

Step 2: Find the GCF.

• GCF of 12 and 18 is 6.

Step 3: Divide both terms by 6.

• 12 ÷ 6 : 18 ÷ 6 = 2:3

Interpretation:

• For every 2 scoops of powder, there are 3 cups of water.

Conclusion: The simplified ratio is 2:3.

Worked example 6: deciding whether a ratio is equivalent without fully simplifying both

Is 14:21 equivalent to 18:27?

We can simplify each or notice the same scaling pattern.

Method 1: simplify each

• 14:21 divide both terms by 7 -> 2:3
• 18:27 divide both terms by 9 -> 2:3

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

Method 2: see the multiplicative structure

• 14:21 is 2 x 7 : 3 x 7
• 18:27 is 2 x 9 : 3 x 9

Both are copies of 2:3.

Conclusion: Yes, the ratios are equivalent.

Common misconceptions and repairs

Misconception 1: adding the same number to both terms keeps the ratio equivalent

Example of the error:

• from 2:3, a student writes 3:4

Why it is wrong:

• 2:3 and 3:4 are not the same comparison.
• 2/3 is not equal to 3/4.

Repair:

• Equivalent ratios come from multiplying or dividing, not adding or subtracting.
• Check by simplifying or comparing as fractions.

Misconception 2: dividing only one term is allowed

Example of the error:

• 12:18 -> 2:18

Why it is wrong:

• Changing only one part changes the comparison.

Repair:

• If you scale one term, you must scale the other by the same non-zero factor.

Misconception 3: a ratio can be simplified by any common-looking number

Example of the error:

• 15:25 -> 4:6

Why it is wrong:

• 15 and 25 were not both divided by the same factor to get 4 and 6.

Repair:

• State the factor aloud: "I am dividing both terms by 5." If you cannot name one shared factor, the step is invalid.

Misconception 4: 0 can be used as a scaling factor

Why it is wrong:

• Multiplying both terms by 0 destroys the original comparison.
• Division by 0 is undefined.

Repair:

• The factor must be non-zero.

Quick checks for reasonableness

When you finish, ask:

• Did I multiply or divide both terms by the same number?
• If I simplified, are the new terms smaller but still in the same comparison?
• Can I check by simplifying both ratios to the same lowest terms?

Practice

Try these on your own before checking the answers.

1. Write two equivalent ratios for 4:9.
2. Simplify 21:35.
3. Are 8:12 and 14:21 equivalent?
4. Find x in 5:6 = x:24.
5. A class has 16 girls and 20 boys. Write the ratio of girls to boys in simplest form.
6. Simplify 45:60, then explain what the simplified ratio means.

Practice answers

1. Possible answers: 8:18 and 12:27.
2. 21:35 = 3:5.
3. Yes. 8:12 = 2:3 and 14:21 = 2:3.
4. x = 20 because 24 ÷ 6 = 4, so 5 x 4 = 20.
5. 16:20 = 4:5.
6. 45:60 = 3:4. This means for every 3 of the first quantity, there are 4 of the second quantity.

Summary

Equivalent ratios describe the same multiplicative comparison. You create them by multiplying or dividing both terms by the same non-zero factor. Simplifying a ratio means finding an equivalent ratio in lowest terms, usually by dividing by the greatest common factor.

What this enables next

This lesson supports later work with:

• comparing ratios efficiently,
• solving proportion-style missing-value problems,
• understanding unit rates and best-buy reasoning,
• checking whether two situations are proportional.

For broader progression, return to the unit overview records. For formal assessment after practice, use the linked unit mastery quiz.