4. Proportions

What Is a Proportion?

A proportion is a statement that two ratios or rates are equivalent.

For example:

2 : 3 = 4 : 6

Both ratios describe the same relationship because:

2 × 2 = 4

and:

3 × 2 = 6

We can also write the proportion using fractions:

2/3 = 4/6

A proportional relationship keeps the same relative relationship between two quantities as the quantities increase or decrease.

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Ratios, Rates, and Proportions

These ideas are closely connected.

A ratio compares quantities.

Example:

2 : 5

A rate usually compares quantities with different units.

Example:

120 km : 2 h

A proportion states that two ratios or rates are equivalent.

Example:

2/5 = 6/15

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Equivalent Ratios Create Proportions

Consider:

3 : 4

Multiply both terms by 2:

6 : 8

Multiply both terms by 3:

9 : 12

Multiply both terms by 5:

15 : 20

Therefore:

3 : 4 = 6 : 8 = 9 : 12 = 15 : 20

Each pair of equivalent ratios can form a proportion.

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What Is a Proportional Relationship?

Two quantities have a proportional relationship when they change by the same scale factor and maintain a constant ratio.

Suppose notebooks cost $3 each.

Then:

1 notebook → $3
2 notebooks → $6
3 notebooks → $9
4 notebooks → $12
5 notebooks → $15

The ratio:

cost / number of notebooks

is always:

3

Therefore, cost and number of notebooks are proportional.

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The Constant of Proportionality

A proportional relationship can be written:

y = kx

where:

k

is the constant of proportionality.

For example:

If notebooks cost $3 each:

cost = 3 × number of notebooks

or:

y = 3x

Therefore:

k = 3

The constant of proportionality is also the unit rate.

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Identifying Proportional Relationships

Suppose we have:

Calculate:

cost / tickets

For each row:

5/1 = 5

10/2 = 5

15/3 = 5

20/4 = 5

The ratio is constant.

Therefore, the relationship is proportional.

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Identifying a Non-Proportional Relationship

Consider:

Check:

7/1 = 7

11/2 = 5.5

15/3 = 5

19/4 = 4.75

The ratios are not constant.

Therefore, this is not a proportional relationship.

There may be another pattern, but it is not proportional.

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Why "Same Increase" Does Not Mean Proportional

In the previous example, the cost increases by:

$4 each time

That is a constant additive change.

However, proportional relationships require a constant multiplicative relationship.

This is an important distinction.

Proportional relationships follow:

y = kx

not:

y = kx + b

when b ≠ 0.

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Proportions Using Fractions

A proportion can be written:

a/b = c/d

For example:

3/5 = 12/20

Both fractions simplify to:

3/5

Therefore, they form a proportion.

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Solving Proportions by Scaling

Suppose:

3/5 = x/20

Ask:

What multiplies 5 to make 20?

5 × 4 = 20

Apply the same scale factor to the numerator:

3 × 4 = 12

Therefore:

x = 12

So:

3/5 = 12/20

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Finding a Missing Denominator

Solve:

4/7 = 12/x

Ask:

What multiplies 4 to make 12?

× 3

Apply the same factor to 7:

7 × 3 = 21

Therefore:

x = 21

Check:

4/7 = 12/21

Both simplify to:

4/7

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Finding a Missing Value by Simplifying

Solve:

x/24 = 3/8

We know:

8 × 3 = 24

Therefore:

3 × 3 = 9

So:

x = 9

Check:

9/24 = 3/8

Correct.

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Solving with a Unit Rate

Suppose:

5 notebooks cost $15

How much do 8 notebooks cost?

First find the unit rate:

$15 ÷ 5 = $3 per notebook

Then:

8 × $3 = $24

Therefore:

8 notebooks cost $24

Unit rates are often one of the easiest ways to solve practical proportions.

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Using Ratio Tables

Suppose:

3 kg of apples cost $12

We can build a ratio table:

The table helps us see equivalent ratios.

For example:

3 kg : $12 = 5 kg : $20

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Scaling Up in a Ratio Table

Suppose a recipe uses:

2 cups flour for 3 batches

How much flour is needed for 12 batches?

Start:

2 cups : 3 batches

Since:

3 × 4 = 12

multiply both quantities by 4:

2 × 4 = 8

Therefore:

8 cups of flour

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Scaling Down

Suppose:

12 notebooks cost $30

What is the cost of 4 notebooks?

Since:

12 ÷ 3 = 4

divide the cost by 3:

$30 ÷ 3 = $10

Therefore:

4 notebooks cost $10

Proportional reasoning works when scaling quantities up or down.

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Double Number Lines

A double number line can represent a proportional relationship.

Suppose:

2 kg → $6

Then:

1 kg → $3
2 kg → $6
3 kg → $9
4 kg → $12
5 kg → $15

The corresponding values remain aligned because the unit rate stays constant.

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Bar Models and Proportions

Suppose:

boys : girls = 2 : 3

and there are:

12 boys

If 2 ratio parts represent 12:

1 part = 12 ÷ 2 = 6

Girls:

3 × 6 = 18

Therefore:

12 boys : 18 girls

simplifies to:

2 : 3

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Cross Multiplication

Another method for solving proportions is cross multiplication.

Suppose:

3/5 = x/20

Cross multiply:

3 × 20 = 5 × x

Therefore:

60 = 5x

Divide by 5:

x = 12

This gives the same answer as scaling.

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Why Cross Multiplication Works

Start with:

a/b = c/d

Multiply both sides by:

bd

Then:

ad = bc

This creates the cross-product rule.

Therefore, if:

a/b = c/d

then:

a × d = b × c

Cross multiplication is useful, but it is important to understand that it comes from equivalent ratios rather than treating it as an unexplained trick.

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Checking Whether Two Ratios Form a Proportion

Do:

4 : 6

and:

10 : 15

form a proportion?

Write:

4/6 = 10/15

Cross products:

4 × 15 = 60

6 × 10 = 60

The cross products are equal.

Therefore:

4/6 = 10/15

and the ratios form a proportion.

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A Non-Proportion Example

Compare:

3/4

and:

8/10

Cross products:

3 × 10 = 30

4 × 8 = 32

Since:

30 ≠ 32

the ratios are not equivalent.

Therefore, they do not form a proportion.

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Proportions in Recipes

A recipe uses:

2 cups of rice for 5 people

How much rice is needed for 20 people?

Set up:

2/5 = x/20

Since:

5 × 4 = 20

calculate:

2 × 4 = 8

Therefore:

8 cups of rice

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Proportions in Maps

A map scale states:

1 cm : 8 km

Two locations are:

6 cm

apart on the map.

Set up:

1 cm / 8 km = 6 cm / x km

Scale:

1 → 6 = ×6

Therefore:

8 × 6 = 48

The actual distance is:

48 km

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Proportions in Scale Drawings

A model uses a scale:

1 : 25

A part of the model measures:

8 cm

Actual measurement:

8 × 25 = 200 cm

Therefore:

200 cm = 2 m

Scale drawings are based on proportional relationships.

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Proportions and Speed

Suppose a vehicle travels:

180 km in 3 hours

At the same average rate, how far would it travel in 5 hours?

First find the unit rate:

180 ÷ 3 = 60 km/h

Then:

60 × 5 = 300 km

Therefore:

300 km

We can also write:

180/3 = x/5

Solving gives:

x = 300

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Proportions and Cost

Suppose:

4 tickets cost $30

How much would 10 tickets cost at the same rate?

Unit price:

$30 ÷ 4 = $7.50

Then:

10 × $7.50 = $75

Therefore:

10 tickets cost $75

This assumes there are no fixed fees, bulk discounts, or other changes to the pricing structure.

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Proportions and Currency-Like Conversions

Suppose an illustrative conversion rate is:

1 unit A = 1.5 units B

Then:

4 units A = 6 units B

because:

4 × 1.5 = 6

This is proportional as long as the conversion rate remains constant.

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Proportions and Percentages

Percentages are proportions based on 100.

Suppose:

18 out of 24 students

completed an assignment.

Set up:

18/24 = x/100

Simplify:

18/24 = 3/4

And:

3/4 = 75/100

Therefore:

x = 75

So:

75%

completed the assignment.

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Finding a Percentage Using a Proportion

What percentage of 80 is 28?

Set up:

28/80 = x/100

Cross multiply:

28 × 100 = 80x

2800 = 80x

x = 35

Therefore:

28 is 35% of 80

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Finding a Quantity Using a Percentage Proportion

Find 30% of 70.

Set up:

30/100 = x/70

Cross multiply:

30 × 70 = 100x

2100 = 100x

x = 21

Therefore:

30% of 70 = 21

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Proportions in Similar Shapes

Proportions are important in geometry.

If two shapes are similar, corresponding side lengths have equal ratios.

For example:

Small triangle:

3 cm, 4 cm, 5 cm

Larger similar triangle:

6 cm, 8 cm, 10 cm

Each length has been multiplied by:

2

Therefore:

3/6 = 4/8 = 5/10

The corresponding sides are proportional.

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Proportions in Science

Proportional reasoning appears throughout science.

Examples include:

• distance and time at constant speed
• mass and volume at constant density
• electrical relationships under suitable conditions
• scale models
• concentrations
• mixtures
• experimental measurements

For example, if a substance has constant density:

density = mass/volume

then mass is proportional to volume.

If the volume doubles, the mass doubles.

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Direct Proportion

A direct proportion occurs when one quantity increases or decreases by the same factor as another.

For example:

If one notebook costs $4:

2 notebooks cost $8.

4 notebooks cost $16.

10 notebooks cost $40.

Doubling the number of notebooks doubles the cost.

Tripling the number triples the cost.

This is direct proportion.

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The Graph of a Proportional Relationship

A direct proportional relationship has an equation:

y = kx

Its graph is a straight line passing through:

(0, 0)

For example:

y = 3x

gives:

x = 0 → y = 0
x = 1 → y = 3
x = 2 → y = 6
x = 3 → y = 9
x = 4 → y = 12

The constant of proportionality is:

k = 3

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Why Must the Graph Pass Through the Origin?

If:

x = 0

then:

y = k(0) = 0

Therefore, every direct proportional relationship contains:

(0, 0)

Suppose a taxi charges:

$5 starting fee + $2 per kilometre

Then:

cost = 2d + 5

At 0 km:

cost = $5

The graph does not pass through the origin.

Therefore, total cost is not directly proportional to distance, even though the cost increases at a constant rate after the starting fee.

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Constant Difference vs Constant Ratio

Consider:

2 → 5
4 → 7
6 → 9
8 → 11

The second quantity increases by 2 each time.

But:

5/2 ≠ 7/4 ≠ 9/6

So the relationship is not proportional.

A proportional relationship requires a constant ratio, not merely a constant difference.

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Proportional Reasoning Without Formal Equations

Suppose:

6 bottles contain 9 L

How much do 2 bottles contain if the bottles are equal-sized?

Since:

6 → 2

means divide by 3:

9 L ÷ 3 = 3 L

Therefore:

2 bottles contain 3 L

This is proportional reasoning even without writing an equation.

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Choosing the Best Method

Different proportion problems may be easier with different methods.

Use scaling when the scale factor is obvious.

Example:

3 : 5 = 12 : 20

Use a unit rate when finding the value for one makes the problem easier.

Example:

$18 for 6 → $3 for 1.

Use a ratio table when several equivalent values are useful.

Use a bar model or double number line when a visual representation helps.

Use cross multiplication when the scale factor is not obvious.

Strong proportional reasoning means choosing an efficient method rather than relying on only one technique.

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Worked Example 1

Solve:

5/8 = x/40

Since:

8 × 5 = 40

calculate:

5 × 5 = 25

Therefore:

x = 25

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Worked Example 2

Solve:

7/12 = 21/x

Since:

7 × 3 = 21

calculate:

12 × 3 = 36

Therefore:

x = 36

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Worked Example 3

Solve:

9/15 = x/35

Simplify:

9/15 = 3/5

Now:

3/5 = x/35

Since:

5 × 7 = 35

calculate:

3 × 7 = 21

Therefore:

x = 21

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Worked Example 4

A printer produces:

180 pages in 6 minutes

How many pages will it produce in 15 minutes at the same rate?

Unit rate:

180 ÷ 6 = 30 pages/min

Then:

30 × 15 = 450 pages

Answer:

450 pages

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Worked Example 5

A recipe for 4 people requires:

300 g of pasta

How much pasta is needed for 10 people?

Unit amount:

300 ÷ 4 = 75 g/person

Then:

75 × 10 = 750 g

Answer:

750 g

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Worked Example 6

A map uses:

2 cm : 15 km

A road measures:

7 cm

on the map.

Set up:

2/15 = 7/x

Cross multiply:

2x = 105

x = 52.5

Therefore:

actual distance = 52.5 km

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Worked Example 7

Determine whether the relationship is proportional.

Calculate:

8/2 = 4

16/4 = 4

24/6 = 4

40/10 = 4

The ratio is constant.

Therefore:

y = 4x

and the relationship is proportional.

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Worked Example 8

Determine whether this relationship is proportional.

Check:

4/1 = 4

6/2 = 3

8/3 ≈ 2.67

10/4 = 2.5

The ratio is not constant.

Therefore, the relationship is not proportional.

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Worked Example 9: Practical Comparison

Machine A produces:

120 items in 4 hours

Machine B produces:

175 items in 5 hours

Machine A:

120 ÷ 4 = 30 items/hour

Machine B:

175 ÷ 5 = 35 items/hour

Using unit rates makes the production rates directly comparable.

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Worked Example 10: Multi-Step Proportion

A drink mixture uses:

concentrate : water = 2 : 7

You want to make:

27 L

of drink.

Total ratio parts:

2 + 7 = 9

One part:

27 ÷ 9 = 3 L

Concentrate:

2 × 3 = 6 L

Water:

7 × 3 = 21 L

Check:

6 + 21 = 27 L

and:

6 : 21 = 2 : 7

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Checking a Proportion

After solving, verify that the two ratios are equivalent.

Suppose:

4/7 = 12/21

Simplify:

12/21 = 4/7

Correct.

Or use cross products:

4 × 21 = 84

7 × 12 = 84

Therefore, the proportion is correct.

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Checking Reasonableness

Suppose:

5 notebooks cost $20

How much should 15 notebooks cost?

15 notebooks is:

3 times

as many notebooks.

Therefore, the cost should also be:

3 times

as large.

$20 × 3 = $60

If a calculation produced $6 or $600, the answer would clearly be unreasonable.

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Common Mistakes

Mistake 1: Changing only one part of a ratio

Incorrect:

2 : 3 → 4 : 3

Correct:

2 : 3 → 4 : 6

Both quantities must scale by the same factor.

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Mistake 2: Mixing up corresponding quantities

If comparing kilograms with dollars, keep kilograms aligned with kilograms and dollars aligned with dollars.

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Mistake 3: Assuming every increasing relationship is proportional

A relationship can increase without maintaining a constant ratio.

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Mistake 4: Looking only for a constant difference

Proportional relationships require a constant ratio.

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Mistake 5: Assuming every straight-line relationship is proportional

A proportional graph must be a straight line through the origin.

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Mistake 6: Cross multiplying incorrectly

For:

a/b = c/d

the correct relationship is:

ad = bc

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Mistake 7: Using proportional reasoning when the rate changes

If prices include fixed fees, changing discounts, or different rates at different levels, the relationship may not be proportional.

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Mistake 8: Forgetting units

A practical answer should include the appropriate unit:

$24

48 km

750 g

rather than only the numerical value.

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Did You Know?

Proportional reasoning is one of the most widely used mathematical ideas.

It is used in:

• recipes
• maps
• scale drawings
• shopping
• currency conversions
• percentages
• speed calculations
• science experiments
• chemistry
• engineering
• construction
• photography
• similar shapes
• probability
• data analysis

Proportional reasoning also provides an important bridge from arithmetic into algebra.

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Key Terms

• Proportion: Statement that two ratios or rates are equivalent.
• Proportional relationship: Relationship in which two quantities maintain a constant ratio.
• Equivalent ratios: Ratios describing the same proportional relationship.
• Scale factor: Number used to multiply or divide corresponding quantities.
• Unit rate: Rate expressed per one unit.
• Constant of proportionality: Constant ratio connecting two proportional quantities.
• Ratio table: Table containing equivalent ratios.
• Double number line: Visual representation of two proportional quantities.
• Bar model: Diagram showing quantities as proportional parts.
• Cross multiplication: Method for solving or checking proportions using cross products.
• Cross product: Product obtained by multiplying diagonally across two ratios written as fractions.
• Direct proportion: Relationship in which one quantity changes by the same factor as another.
• Origin: Point (0, 0) on a coordinate graph.

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Key Equations and Rules

A proportion can be written:

a/b = c/d

For equivalent ratios:

ad = bc

A direct proportional relationship can be written:

y = kx

where:

k = y/x

and k is the constant of proportionality.

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Proportion Problem-Solving Strategy

Step 1: Identify the two related quantities.

Step 2: Decide whether the relationship is proportional.

Step 3: Keep corresponding quantities in the same order.

Step 4: Write equivalent ratios or create a table or diagram.

Step 5: Look for an easy scale factor.

Step 6: If useful, find the unit rate.

Step 7: If necessary, use cross multiplication.

Step 8: Calculate the missing value.

Step 9: Include the correct units.

Step 10: Verify that the ratios are equivalent and the answer is reasonable.

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Key Takeaways

• A proportion states that two ratios or rates are equivalent.
• Proportional relationships maintain a constant ratio.
• Equivalent ratios can be created by multiplying or dividing corresponding quantities by the same non-zero scale factor.
• Missing values can often be found by scaling up or down.
• Unit rates provide another powerful way to solve proportional problems.
• Ratio tables organize equivalent ratios.
• Bar models and double number lines provide visual representations of proportional relationships.
• Cross multiplication can solve proportions when the scale factor is not obvious.
• Cross multiplication works because the ratios are equivalent.
• In a proportional relationship, y/x remains constant.
• The constant ratio is called the constant of proportionality.
• Direct proportional relationships can be represented by y = kx.
• A graph of a direct proportional relationship is a straight line through the origin.
• A constant additive change does not automatically mean a relationship is proportional.
• Proportional reasoning should only be used when the relationship actually maintains a constant rate or ratio.
• Proportions are useful in percentages, recipes, maps, scale drawings, speed, shopping, science, and many other practical situations.
• A useful question to ask is:

If one quantity changes by a certain factor, does the other quantity change by the same factor?