Home / Mathematics / Grades 6–7 / Rates and Proportions
GRADES 6–7 · MATHEMATICS

Rates and Proportions

A rate compares quantities measured in different units, while a proportion states that two ratios are equal. Learn how to calculate unit rates, compare value and solve simple proportion problems in everyday situations.

Learning time: approximately 30 minutes · Beginner to developing level

What You Will Learn

1. What Is a Rate?

A rate compares two quantities measured in different units. Examples include kilometres per hour, dollars per kilogram and pages per minute.

Example: A cyclist travels 30 kilometres in 2 hours. The rate is 30 kilometres per 2 hours.

A ratio can compare quantities with the same or different units. A rate is a ratio that compares different units.

2. Find a Unit Rate

A unit rate tells you the amount for one unit. Divide both parts of the rate by the number of units in the second quantity.

Worked example: 30 kilometres in 2 hours

  1. Write the rate: 30 km in 2 hours.
  2. Divide both quantities by 2.
  3. 30 ÷ 2 = 15 km; 2 ÷ 2 = 1 hour.

Unit rate: 15 kilometres per hour (15 km/h).

Another example: Cost per notebook

Four notebooks cost $12. Divide $12 by 4. The unit price is $3 per notebook.

3. Compare Unit Rates

To compare two deals, calculate the unit rate for each deal. Make sure both rates use the same units.

Example: Pack A has 6 pencils for $3. Pack B has 10 pencils for $4.

Pack B costs less per pencil, so it offers better value if the pencil quality and other conditions are the same.

4. What Is a Proportion?

A proportion is a statement that two ratios are equal. For example, 2:3 and 4:6 are equivalent ratios, so they form a proportion.

2 : 3 = 4 : 6

Both terms of 2:3 have been multiplied by 2 to make 4:6.

Equivalent ratios are made by multiplying or dividing every term by the same non-zero number.

5. Solve a Missing-Value Proportion

You can use equivalent ratios to find a missing value.

Worked example: 3 : 5 = 12 : x

  1. Compare the first terms: 3 becomes 12 by multiplying by 4.
  2. Multiply the second term by the same factor: 5 × 4 = 20.
  3. Therefore, x = 20.

Answer: 3:5 = 12:20.

Another example: 2 : 7 = x : 21

7 becomes 21 by multiplying by 3. Multiply 2 by 3 as well. Therefore, x = 6.

6. Solve Everyday Problems

Example 1: Travel speed

A car travels 180 kilometres in 3 hours at a steady average rate. What is the distance per hour?

180 ÷ 3 = 60 km/h.

Example 2: Recipe amounts

A recipe uses 2 cups of rice for 5 people. How many cups are needed for 10 people if the same amount per person is used?

10 is twice 5, so double the rice: 2 × 2 = 4 cups.

Example 3: Printing

A printer makes 45 pages in 3 minutes. At the same rate, how many pages does it make in one minute?

45 ÷ 3 = 15 pages per minute.

7. Common Mistakes to Avoid

Comparing different units: Convert rates to the same unit before deciding which is greater or better value.
Scaling only one term: In an equivalent ratio, multiply or divide both terms by the same non-zero factor.
Mixing up the order: 3 dollars per item is not the same as 3 items per dollar. State the units clearly.
Assuming a rate stays constant: Proportion problems assume the relationship remains constant, as stated in the question.

8. Practice Exercises

Try these before revealing the answers.

  1. A runner travels 12 km in 3 hours. Find the unit rate in km/h.
  2. Five pens cost $10. Find the cost per pen.
  3. Which is cheaper per kilogram: 2 kg for $6 or 3 kg for $7.50?
  4. Complete the proportion: 2 : 5 = 8 : x.
  5. Complete the proportion: 4 : 7 = x : 21.
  6. A machine makes 60 parts in 4 minutes. How many parts does it make per minute?
  7. A drink mix uses 1 cup of concentrate for 4 cups of water. How much water is needed for 3 cups of concentrate?
Show practice answers and working
  1. 12 ÷ 3 = 4 km/h.
  2. $10 ÷ 5 = $2 per pen.
  3. First: $6 ÷ 2 = $3/kg. Second: $7.50 ÷ 3 = $2.50/kg. The 3 kg pack is cheaper per kilogram.
  4. 2 becomes 8 by multiplying by 4; x = 5 × 4 = 20.
  5. 7 becomes 21 by multiplying by 3; x = 4 × 3 = 12.
  6. 60 ÷ 4 = 15 parts per minute.
  7. Multiply both parts by 3: 3 cups concentrate need 12 cups water.

9. Test Your Understanding

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

1. A rate compares quantities measured in...
2. Six oranges cost $3. What is the cost per orange?
3. Which ratio is equivalent to 2 : 3?
4. Complete: 3 : 4 = 9 : x.
5. A car travels 100 km in 2 hours. What is its average distance per hour?

10. Free Printable Worksheet

Print this worksheet or save it as a PDF for extra practice.

Learnora — Grades 6–7 Mathematics

Rates and Proportions Worksheet

Name: __________________________ Date: _______________

Instructions: Show your working and include units.

  1. Eight apples cost $4. Find the cost per apple.
  2. A cyclist travels 42 km in 3 hours. Find the average distance per hour.
  3. Complete: 2 : 7 = 6 : ____.
  4. Complete: 5 : 8 = ____ : 24.
  5. Compare: 4 kg for $12 or 5 kg for $14. Which costs less per kilogram?
  6. A recipe uses 3 cups of flour for 6 people. How much flour is needed for 12 people?

Keep practising — you are making progress!

Learnora — Learn Better. Practice Smarter.

Teacher / Parent Answer Key
  1. $4 ÷ 8 = $0.50 per apple.
  2. 42 ÷ 3 = 14 km/h.
  3. 7 becomes 21 by multiplying by 3, so the missing term is 21.
  4. 8 becomes 24 by multiplying by 3, so the missing term is 15.
  5. First: $12 ÷ 4 = $3/kg. Second: $14 ÷ 5 = $2.80/kg. The 5 kg option costs less per kilogram.
  6. Double 3 cups because 12 is twice 6. Answer: 6 cups.

11. Lesson Summary

Next learning goal: Apply rates and proportions to maps, scale drawings and more complex problems.

Educational note: This is a curriculum-neutral lesson for Grades 6–7. Teachers and parents can adapt examples and practice to their local curriculum.