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Teacher Guide: Equivalent Ratios

Learn how to find and create ratios that represent the same relationship.

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Class quiz

10 questions on Ratios. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Identify equivalent ratios using multiplication and division
  • Generate equivalent ratios from a given ratio
  • Determine if two ratios are equivalent by simplifying
  • Complete ratio tables with missing values
  • Apply equivalent ratios to solve real-world problems
Prerequisites
  • • Understanding of what a ratio represents
  • • Multiplication and division of whole numbers
  • • Concept of equivalent fractions
  • • Finding greatest common factor (GCF)
Discussion Starters
  • 1. If you double both parts of a ratio, does the ratio change? Why or why not?
  • 2. A store sells 3 apples for 2 dollars. What are some equivalent ratios for this? How could you use them?
  • 3. Why does adding the same number to both parts NOT give an equivalent ratio?
  • 4. How could you use equivalent ratios to compare prices at two different stores?
Common Misconceptions

Adding the same number to both parts creates an equivalent ratio

Remediation: Use visual models. Show that 2:3 (2 red squares to 3 blue squares) changes when you add 1 to each (3:4 is a different proportion). Then show multiplying preserves the proportion.

Larger numbers mean the ratio is bigger

Remediation: Compare 2:4 and 100:200. Both simplify to 1:2. Use fraction form: 2/4 = 100/200 = 0.5. The value is the same regardless of the numbers used.

You can only multiply, not divide, to find equivalent ratios

Remediation: Show that 8:12 ÷ 4 = 2:3, and 2:3 × 4 = 8:12. Division finds simpler equivalent ratios; multiplication finds larger ones. Both are valid.

Differentiation Ideas

For Struggling Students:

  • • Use physical manipulatives (colored counters) to show ratios
  • • Start with simple ratios like 1:2 and 2:4
  • • Provide partially completed ratio tables
  • • Use visual tape diagrams to show equivalent relationships

For On-Level Students:

  • • Complete ratio tables with multiple missing values
  • • Determine if given ratios are equivalent
  • • Solve word problems involving equivalent ratios
  • • Create their own equivalent ratio problems

For Advanced Students:

  • • Work with three-part ratios (2:3:5)
  • • Explore ratios with fractions or decimals
  • • Solve complex scaling problems (triple the recipe, then halve it)
  • • Connect equivalent ratios to proportional reasoning
Standards Alignment
  • 6.RP.A.1 (CCSS.MATH.CONTENT.6.RP.A.1)

    Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities

  • 6.RP.A.3 (CCSS.MATH.CONTENT.6.RP.A.3)

    Use ratio and rate reasoning to solve real-world and mathematical problems

  • 6.RP.A.3.A (CCSS.MATH.CONTENT.6.RP.A.3.A)

    Make tables of equivalent ratios relating quantities with whole number measurements

Lesson Resources
  • visualInteractive Ratio Table

    Students fill in missing values in ratio tables

  • activityRecipe Scaling Game

    Adjust recipe ingredients for different serving sizes

  • worksheetEquivalent Ratios Practice

    Find missing values and determine equivalence

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

Equivalent ratios are two or more ratios that express the same relationship between quantities. Just like equivalent fractions, equivalent ratios have the same value when simplified.
You can find equivalent ratios by multiplying or dividing both parts of a ratio by the same number:
2:3=4:6=6:9=8:122:3 = 4:6 = 6:9 = 8:12
All these ratios are equivalent because they all simplify to 2:32:3.

Worked Examples

Find three ratios equivalent to 2:52:5.
1

Multiply both parts by 2

2×2=42 \times 2 = 4 and 5×2=105 \times 2 = 10 → 4:104:10
2

Multiply both parts by 3

2×3=62 \times 3 = 6 and 5×3=155 \times 3 = 15 → 6:156:15
3

Multiply both parts by 4

2×4=82 \times 4 = 8 and 5×4=205 \times 4 = 20 → 8:208:20

Common Mistakes

Adding the same number to both parts instead of multiplying

Why it's wrong: Adding changes the ratio's value. If 2:32:3 becomes 4:54:5 (adding 2), these are NOT equivalent.

Correct: Always multiply or divide both parts by the SAME number. 2:3=4:62:3 = 4:6 (multiplying by 2).

Multiplying only one part of the ratio

Why it's wrong: To keep ratios equivalent, you must apply the same operation to BOTH parts.

Correct: 2:5×3=6:152:5 \times 3 = 6:15, not 6:56:5 or 2:152:15.

Thinking larger numbers mean larger ratios

Why it's wrong: 4:64:6 looks bigger than 2:32:3, but they represent the same relationship.

Correct: The size of the numbers does not determine the ratio's value. Always simplify to compare.

Why It Matters

Equivalent ratios help us solve real-world problems:
  • Cooking: If a recipe calls for 2 cups of flour to 3 cups of sugar, you can double it to 4 cups of flour and 6 cups of sugar
  • Maps: A scale of 1:100 means 1 cm on the map equals 100 cm in real life
  • Mixing: If paint requires a 3:1 ratio of white to blue, you need to keep this ratio no matter how much paint you make
  • Shopping: Comparing unit prices uses equivalent ratios to find the best deal
Understanding equivalent ratios is the foundation for solving proportions!

Real World Applications

Scaling Recipes

Chefs use equivalent ratios to adjust recipes for different numbers of servings.

Example:

A recipe for 4 people uses 3 eggs and 2 cups of flour. For 8 people, use 6 eggs and 4 cups of flour (multiply by 2).

1Try It Yourself

A smoothie recipe uses 2 bananas for every 3 cups of milk. You want to make enough for 6 bananas.

How many cups of milk do you need?

Step 1: Write the mathematical expression

If 2 bananas need 3 cups, find the multiplier:

Map Scales

Maps use ratios to represent real distances. A scale of 1:50,000 means 1 cm on the map equals 50,000 cm (or 500 m) in reality.

Example:

On a 1:100 scale map, 5 cm represents 500 cm (5 meters) in real life.

2Try It Yourself

A map has a scale of 1:25,000. Two cities are 8 cm apart on the map.

What is the real distance in kilometers?

Step 1: Write the mathematical expression

Calculate the real distance:

Mixing Paint

Artists mix colors using specific ratios to get consistent shades.

Example:

To make light green, mix blue and yellow in a 1:4 ratio. For 20 ml total, use 4 ml blue and 16 ml yellow.

3Try It Yourself

A shade of orange requires red and yellow paint in a 2:5 ratio. You have 10 ml of red paint.

How much yellow paint do you need?

Step 1: Write the mathematical expression

Find the equivalent ratio:

Key Takeaways

  • 1Equivalent ratios express the same relationship between quantities
  • 2To create equivalent ratios, multiply or divide BOTH parts by the same number
  • 3You can check if ratios are equivalent by simplifying them to lowest terms
  • 4Ratio tables help organize and find multiple equivalent ratios
  • 5Never add or subtract to find equivalent ratios - only multiply or divide

Frequently Asked Questions

How are equivalent ratios different from equivalent fractions?

They work the same way! A ratio 2:32:3 can be written as the fraction 23\frac{2}{3}. Finding equivalent ratios is just like finding equivalent fractions - multiply or divide both parts by the same number.

Can ratios have decimals?

Yes, but we usually simplify them to whole numbers. For example, 1.5:31.5:3 is equivalent to 1:21:2 (divide both by 1.5) or 3:63:6 (multiply both by 2).

What is the simplest form of a ratio?

A ratio is in simplest form when both parts share no common factors other than 1. For example, 6:96:9 simplifies to 2:32:3 by dividing both by their GCF (3).

Glossary

Equivalent ratios
Two or more ratios that represent the same relationship between quantities
Ratio table
A table showing pairs of numbers that all form equivalent ratios
Simplest form
A ratio where both terms have no common factors other than 1
Scale factor
The number you multiply both parts of a ratio by to get an equivalent ratio

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