GRADE 7 · MATHEMATICS

Ratios, Rates and Proportional Relationships

Ratios help us compare quantities. Rates compare quantities measured in different units, and proportional relationships describe situations where two quantities change at a constant rate. Learn to represent, compare and solve these relationships using clear methods and everyday examples.

Learning time: approximately 35 minutes · Developing level · Includes practice, quiz and printable worksheet

What You Will Learn

Write and interpret ratios in different forms.
Distinguish part-to-part and part-to-whole comparisons.
Simplify ratios and generate equivalent ratios.
Use ratio tables to find missing values.
Calculate and compare unit rates.
Recognise proportional relationships and their constant of proportionality.
Solve real-world problems involving recipes, prices, distance and time.

1. Understanding Ratios

A ratio compares two quantities by showing how much of one there is in relation to the other. A ratio can be written using a colon, the word “to”, or a fraction bar.

2 : 32 to 32/3

These are three ways to write the same comparison. The context tells you what each number represents.

Example: Red and blue counters

A box contains 4 red counters and 6 blue counters.

Red to blue: 4:6.
Blue to red: 6:4.
Red to all counters: 4:10.
Blue to all counters: 6:10.

Important: The order matters. Red to blue is not the same comparison as blue to red.

Part-to-part and part-to-whole

A part-to-part ratio compares one group with another group. A part-to-whole ratio compares one group with the total. In the example, 4:6 is red to blue, while 4:10 is red to all counters.

2. Simplifying Ratios

You simplify a ratio by dividing every term by the same common factor. This keeps the comparison equivalent while using smaller whole numbers.

Worked example: Simplify 12:18.

The greatest common factor of 12 and 18 is 6.
Divide both terms by 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3.
So 12:18 simplifies to 2:3.

Check by scaling the simplified ratio back up: 2 × 6 = 12 and 3 × 6 = 18.

Common mistake: Dividing only one term changes the comparison. Always apply the same non-zero factor to every term.

3. Equivalent Ratios

Equivalent ratios describe the same relationship. Multiply or divide every term by the same non-zero number to create an equivalent ratio.

Start with 2:5.

Multiply both terms by 2 → 4:10.
Multiply both terms by 3 → 6:15.
Multiply both terms by 4 → 8:20.

All four ratios are equivalent.

Use a ratio table

A ratio table organises equivalent pairs and makes patterns easier to see.

First quantity	2	4	6	8
Second quantity	5	10	15	20

Each column keeps the same relationship: the second quantity is 2.5 times the first.

4. What Is a Rate?

A rate is a ratio that compares quantities with different units. Examples include kilometres per hour, dollars per kilogram and pages per minute.

Example: A cyclist travels 36 kilometres in 3 hours. The rate is 36 kilometres per 3 hours. The unit rate is 36 ÷ 3 = 12 kilometres per hour (12 km/h).

A unit rate tells you the amount for one unit of the second quantity. To find it, divide both parts by the second quantity.

Example: Price per item

Five notebooks cost $15. Divide $15 by 5 notebooks: the unit price is $3 per notebook.

5. Compare Rates Fairly

To compare two deals, convert both to the same unit rate. Do not compare package totals when the package sizes are different.

Example: Which fruit pack is better value?

Pack A: 2 kg for $6 → $6 ÷ 2 = $3 per kg.
Pack B: 3 kg for $7.50 → $7.50 ÷ 3 = $2.50 per kg.

Pack B has the lower cost per kilogram, assuming quality and other conditions are the same.

Always include units in your answer. “2.5” alone does not tell the reader whether the rate is dollars per kilogram, kilograms per dollar or something else.

6. Proportional Relationships

Two quantities have a proportional relationship when one is always a constant multiple of the other. If the first quantity is multiplied by a number, the second is multiplied by the same number.

For a proportional relationship, the ratio between corresponding values stays constant. In the equation y = kx, the number k is called the constant of proportionality.

Example: Buying identical notebooks for $3 each.

If x is the number of notebooks and y is the total cost in dollars, then y = 3x. The constant of proportionality is k = 3 dollars per notebook.

Notebooks (x)	1	2	3	4
Cost ($) (y)	3	6	9	12

Divide cost by notebook count in every column: 3 ÷ 1 = 6 ÷ 2 = 9 ÷ 3 = 12 ÷ 4 = 3. The constant stays 3.

How to check a table

Pair each x-value with its matching y-value.
Calculate y ÷ x for each pair (when x is not zero).
If the quotient is the same for every pair, the table represents a proportional relationship.

Not every relationship is proportional. If a taxi charges a fixed starting fee plus a cost per kilometre, the total cost does not start at zero and may not be proportional to distance.

7. Solve Proportion Problems

A proportion states that two ratios are equal. You can solve a missing value by finding the scale factor between corresponding terms.

Worked example: 3:4 = 12:x

3 becomes 12 by multiplying by 4.
Multiply the second term by the same factor: 4 × 4 = 16.
Therefore, x = 16.

Worked example: 5:2 = x:8

2 becomes 8 by multiplying by 4.
Multiply 5 by 4 as well: 5 × 4 = 20.
Therefore, x = 20.

Keep the order of the quantities consistent. If the first ratio is red counters to blue counters, the second ratio must use the same order.

8. Ratios and Rates in Everyday Life

Recipe scaling

A recipe uses 2 cups of flour for 3 batches of muffins. How much flour is needed for 6 batches? Since 6 is twice 3, double the flour: 2 × 2 = 4 cups.

Travel

A bus travels 150 km in 3 hours at a steady average rate. Its average rate is 150 ÷ 3 = 50 km/h. At that same rate, in 5 hours it would travel 50 × 5 = 250 km.

Map scale

A map uses a scale of 1 cm to represent 5 km. A route measuring 4 cm on the map represents 4 × 5 = 20 km in real distance, assuming the map scale is consistent.

Classroom mixture

The ratio of blue paint to white paint is 2:3. If you use 8 cups of blue paint, multiply both parts of the ratio by 4. You need 12 cups of white paint.

9. Common Mistakes to Avoid

Reversing the order: 2:5 and 5:2 describe different comparisons. Label each quantity before writing the ratio.

Changing only one term: Equivalent ratios require the same multiplier or divisor for every term.

Forgetting units: Write dollars per item, kilometres per hour or another appropriate unit with a rate.

Assuming every situation is proportional: Look for a constant multiplier and check whether the relationship has a zero starting value.

Comparing unequal packages by total price: Calculate the price per one unit first.

10. Practice Exercises

Try each question independently before opening the answer key.

A basket has 8 green apples and 12 red apples. Write the ratio of green to red apples and simplify it.
There are 6 boys and 9 girls in a group. Write the ratio of girls to all students in simplest form.
Complete the equivalent ratio: 3:7 = 12:__.
Seven pens cost $14. Find the cost per pen.
Which is cheaper per litre: 2 litres for $5.40 or 3 litres for $7.50?
A printer produces 72 pages in 4 minutes. Find its unit rate in pages per minute.
If y = 4x, find y when x = 6.
A recipe needs 3 cups of rice for 4 people. How many cups are needed for 12 people at the same rate?

Show practice answers and working

8:12. Divide both terms by 4: 2:3.
There are 15 students in total. Girls to all students = 9:15 = 3:5.
3 becomes 12 by multiplying by 4, so 7 × 4 = 28.
$14 ÷ 7 = $2 per pen.
First: $5.40 ÷ 2 = $2.70/L. Second: $7.50 ÷ 3 = $2.50/L. The 3-litre option is cheaper per litre.
72 ÷ 4 = 18 pages per minute.
y = 4 × 6 = 24.
12 is 3 times 4, so 3 × 3 = 9 cups.

12. Free Printable Worksheet

Print this worksheet or save it as a PDF. Show your working and include units where needed.

Learnora — Grade 7 Mathematics