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GRADES 6–7 · MATHEMATICS

Introduction to Ratios

Learn how a ratio compares two or more quantities. Explore simple pictures, write ratios in different forms, simplify them and solve everyday problems.

Learning time: approximately 25–30 minutes · Beginner level

What You Will Learn

1. What Is a Ratio?

A ratio is a way to compare two or more quantities. It tells us how much of one quantity there is compared with another.

Imagine a fruit bowl containing 3 apples and 2 oranges. The ratio of apples to oranges is 3 to 2.

3 apples compared with 2 oranges
🍎 🍎 🍎3 apples
🍊 🍊2 oranges

Apples : Oranges = 3 : 2

Ratios are used in recipes, maps, classroom groups, sports and many other everyday situations.

2. How to Write a Ratio

The ratio of 3 apples to 2 oranges can be written in three common ways:

Important: The order matters. Apples to oranges is 3 : 2, but oranges to apples is 2 : 3. Always read the quantities in the same order as the question.

3. Ratios with More Than Two Quantities

A ratio can compare three or more quantities. Suppose a class activity uses 2 red counters, 4 blue counters and 3 yellow counters.

2Red
4Blue
3Yellow
Red : Blue : Yellow = 2 : 4 : 3

Keep the quantities in the order given. The ratio has three parts because it compares three groups.

4. Simplifying Ratios

A ratio is in its simplest form when its terms have no common factor greater than 1.

Worked example: Simplify 8 : 12

  1. Find a number that divides both 8 and 12 exactly.
  2. The greatest common factor is 4.
  3. Divide both terms by 4: 8 ÷ 4 = 2 and 12 ÷ 4 = 3.
  4. The simplified ratio is 2 : 3.
Check: 2 : 3 and 8 : 12 are equivalent because multiplying both 2 and 3 by 4 gives 8 and 12.

Another example: Simplify 15 : 25

The greatest common factor of 15 and 25 is 5. Divide both terms by 5 to get 3 : 5.

5. Equivalent Ratios

Equivalent ratios have the same comparison, even though their numbers may be different. Multiply or divide every term by the same non-zero number to make an equivalent ratio.

OperationRatioEquivalent ratio
Start2 : 32 : 3
Multiply both terms by 22 : 34 : 6
Multiply both terms by 32 : 36 : 9
Multiply both terms by 42 : 38 : 12
Avoid this mistake: Do not add the same number to both terms and assume the ratio stays equivalent. For example, 2 : 3 is not equivalent to 3 : 4. Multiply or divide both terms by the same factor instead.

6. Solved Everyday Problems

Example 1: Boys and girls

A club has 10 boys and 15 girls. What is the ratio of boys to girls in simplest form?

  1. Write the ratio in the requested order: 10 : 15.
  2. The greatest common factor is 5.
  3. Divide both terms by 5: 2 : 3.

Answer: 2 : 3.

Example 2: Making a drink

A drink is made using 1 cup of concentrate for every 4 cups of water. How much water is needed for 3 cups of concentrate?

  1. The ratio of concentrate to water is 1 : 4.
  2. Three cups of concentrate is 1 × 3.
  3. Multiply the water amount by the same factor: 4 × 3 = 12.

Answer: 12 cups of water.

Example 3: Find a missing term

Complete the equivalent ratio: 3 : 5 = 12 : ?

  1. 3 has been multiplied by 4 to become 12.
  2. Multiply 5 by the same factor: 5 × 4 = 20.

Answer: 20. The completed ratio is 12 : 20.

7. Common Mistakes to Avoid

Mixing up the order: 4 cats to 7 dogs is 4 : 7, not 7 : 4.
Changing only one term: To make an equivalent ratio, multiply or divide every term by the same factor.
Forgetting to simplify: If both terms have a common factor, divide by it to write the ratio in simplest form.
Confusing a ratio with a total: A ratio of 2 : 3 has 5 total parts when describing a whole split into those two groups, but the ratio itself is still 2 : 3.

8. Practice Exercises

Try these before opening the answers.

  1. Write the ratio of 5 green pens to 2 blue pens.
  2. Write the ratio of 4 cats to 9 dogs in the order dogs to cats.
  3. Simplify 6 : 9.
  4. Simplify 14 : 21.
  5. Complete the equivalent ratio: 2 : 5 = 6 : ?
  6. A recipe uses 2 cups of rice for 3 cups of water. How much water is needed for 6 cups of rice?
Show practice answers
  1. 5 : 2.
  2. 9 : 4.
  3. 2 : 3.
  4. 2 : 3.
  5. 15, because both terms are multiplied by 3.
  6. 9 cups of water, because the rice amount is multiplied by 3, so the water amount is also multiplied by 3.

9. Test Your Understanding

Choose one answer for each question, then select Check Answers. Each question is worth one point.

1. A basket has 4 pears and 3 apples. What is the ratio of pears to apples?
2. What is 6 : 9 in simplest form?
3. Which ratio is equivalent to 2 : 5?
4. Complete: 3 : 4 = 9 : ?
5. A group has 8 adults and 12 children. What is the simplified ratio of adults to children?

10. Free Printable Worksheet

Print this worksheet or save it as a PDF to practise ratios away from the screen.

Learnora — Grades 6–7 Mathematics

Introduction to Ratios Worksheet

Name: __________________________ Date: _______________

Instructions: Answer all questions. Show your working where needed.

  1. Write the ratio of 7 red beads to 5 blue beads.
  2. Write the ratio of 3 dogs to 8 cats in the order cats to dogs.
  3. Simplify 12 : 18.
  4. Simplify 20 : 30.
  5. Complete: 4 : 7 = 12 : ____.
  6. A recipe uses 2 parts juice and 5 parts water. How much water is needed for 6 parts juice?
  7. Explain why 3 : 4 and 6 : 8 are equivalent ratios.

Great work! Keep practising.

Learnora — Learn Better. Practice Smarter.

Teacher / Parent Answer Key
  1. 7 : 5.
  2. 8 : 3.
  3. 2 : 3.
  4. 2 : 3.
  5. 21.
  6. 15 parts of water.
  7. Both terms in 3 : 4 have been multiplied by 2 to make 6 : 8.

11. Lesson Summary

Next learning goal: Practise solving more ratio problems and then explore rates and proportions.

Educational note: This is a curriculum-neutral introductory lesson. Teachers and parents can adapt the examples to their local curriculum.