2. Solving Proportions

Solving Proportions

Introduction

Proportions are used to compare quantities that maintain the same relationship. They help us solve problems involving recipes, maps, measurements, prices, scale drawings, and many other everyday situations.

For example, if 3 notebooks cost $12, how much would 8 notebooks cost at the same price per notebook?

We can solve this problem using a proportion.

A proportion is an equation stating that two ratios are equal.

Understanding proportions allows us to calculate missing values, recognise proportional relationships, and check whether quantities are being compared correctly.

What Is a Proportion?

A proportion is an equation showing that two ratios are equivalent.

For example:

2/3 = 4/6

Both fractions represent the same relationship.

The first ratio compares 2 parts to 3 parts.

The second ratio compares 4 parts to 6 parts.

Because both quantities were multiplied by 2, the ratios are equivalent.

Writing Proportions

A proportion can be written using fractions:

a/b = c/d

or using ratio notation:

a : b = c : d

For these expressions, b and d must not equal zero.

Example

3 : 5 = 9 : 15

This can also be written as:

3/5 = 9/15

Important Observation

The order of the quantities must remain consistent in both ratios.

Identifying Proportional Relationships

Two quantities have a proportional relationship when one quantity is always a constant multiple of the other.

For example, suppose pencils cost $2 each.

To determine whether the relationship is proportional, divide the total cost by the number of pencils.

2/1 = 2

4/2 = 2

6/3 = 2

8/4 = 2

10/5 = 2

The ratio is always 2.

Therefore, the quantities are proportional.

General Equation

A proportional relationship can be written as:

y = kx

where:

• x = independent variable

• y = dependent variable

• k = constant of proportionality

For the pencil example:

C = 2n

where C is the total cost and n is the number of pencils.

Important Observation

In a proportional relationship, the ratio y/x remains constant for all nonzero values of x in the domain.

Recognising Proportional Relationships in Tables

Tables can help us determine whether two quantities are proportional.

Example A

Calculate y/x:

6/2 = 3

12/4 = 3

18/6 = 3

24/8 = 3

The ratio is constant.

Therefore:

y = 3x

The relationship is proportional.

Example B

Calculate y/x:

7/2 = 3.5

13/4 = 3.25

19/6 ≈ 3.17

25/8 = 3.125

The ratios are not equal.

Therefore, the relationship is not proportional.

Important Observation

A relationship can increase at a constant rate without being proportional.

For example, y = 3x + 1 is linear but not proportional because it does not pass through the origin.

Recognising Proportions from Graphs

A proportional relationship produces a straight-line graph passing through the origin (0, 0).

Important Characteristics

A proportional graph:

• Is a straight line.

• Passes through the origin.

• Has a constant slope.

• Can be represented by y = kx.

A straight line that does not pass through the origin is not a proportional relationship.

Equivalent Ratios

Equivalent ratios represent the same comparison.

For example:

2 : 5 = 4 : 10 = 6 : 15 = 8 : 20

To create equivalent ratios, multiply or divide both terms by the same nonzero number.

Worked Example

Find an equivalent ratio to 3 : 7 where the first term is 15.

Step 1: Find the Multiplier

15 ÷ 3 = 5

Step 2: Multiply the Second Term

7 × 5 = 35

Final Answer

3 : 7 = 15 : 35

Checking

15/35 simplifies to 3/7.

Therefore, the ratios are equivalent.

Solving Missing-Value Problems Using Equivalent Ratios

A missing value can often be found by identifying the scale factor.

Worked Example

Solve:

4/9 = 20/x

Step 1: Compare the Numerators

4 × 5 = 20

The scale factor is 5.

Step 2: Apply the Same Factor

9 × 5 = 45

Final Answer

x = 45

Verification

4/9 = 20/45

Simplify:

20/45 = 4/9

The solution is correct.

Important Observation

Using equivalent ratios is often the quickest method when the scale factor is easy to recognise.

What Is Cross Multiplication?

Cross multiplication is a method used to solve equations containing two equal fractions.

For the proportion:

a/b = c/d

where b ≠ 0 and d ≠ 0:

a × d = b × c

This is called cross multiplication because the numerator of each fraction is multiplied by the denominator of the other fraction.

Example

3/4 = 6/8

Cross multiply:

3 × 8 = 4 × 6

24 = 24

The products are equal.

Therefore, the proportion is true.

Why Does Cross Multiplication Work?

Starting with:

a/b = c/d

Multiply both sides by bd:

ad = bc

Cross multiplication is therefore a consequence of multiplying both sides of an equation by the same nonzero quantity.

Visualising Cross Multiplication

Solving Proportions Using Cross Multiplication

Worked Example

Solve:

5/8 = x/24

Step 1: Cross Multiply

5 × 24 = 8 × x

120 = 8x

Step 2: Divide Both Sides by 8

x = 120 ÷ 8

x = 15

Step 3: Verify

5/8 = 15/24

Simplify 15/24:

15/24 = 5/8

The ratios are equivalent.

Cross Multiplication with an Unknown Denominator

Worked Example

Solve:

7/x = 14/18

Step 1: Cross Multiply

7 × 18 = 14x

126 = 14x

Step 2: Solve

x = 126 ÷ 14

x = 9

Step 3: Check

7/9 = 14/18

Simplify 14/18:

14/18 = 7/9

The solution is correct.

Important Observation

The unknown may appear in either the numerator or denominator.

The cross-multiplication procedure remains the same, provided the denominators are nonzero.

Cross Multiplication with Expressions

Sometimes the unknown is part of an algebraic expression.

Worked Example

Solve:

(x + 2)/5 = 4/3

Step 1: Cross Multiply

3(x + 2) = 5 × 4

3(x + 2) = 20

Step 2: Expand

3x + 6 = 20

Step 3: Solve

3x = 14

x = 14/3

Step 4: Verify

x + 2 = 14/3 + 6/3 = 20/3

Left side:

(20/3)/5 = 4/3

Right side:

4/3

Both sides are equal.

Solving Proportions with Decimals

Worked Example

Solve:

2.5/4 = x/12

Step 1: Cross Multiply

2.5 × 12 = 4x

30 = 4x

Step 2: Solve

x = 30 ÷ 4

x = 7.5

Verification

2.5/4 = 0.625

7.5/12 = 0.625

The solution is correct.

Important Observation

Proportions can involve whole numbers, fractions, decimals, and algebraic expressions.

Proportions in Recipes

Recipes frequently involve proportional relationships.

Worked Example

A recipe requires 3 cups of flour to make 12 muffins.

How much flour is needed to make 20 muffins, assuming the recipe scales proportionally?

Step 1: Define the Unknown

Let x = cups of flour required.

Step 2: Write the Proportion

3/12 = x/20

Step 3: Cross Multiply

3 × 20 = 12x

60 = 12x

Step 4: Solve

x = 5

Interpretation

The recipe requires 5 cups of flour to make 20 muffins.

Checking

12 muffins require 3 cups.

20 muffins require more flour.

Five cups is reasonable.

Proportions in Maps and Scale Drawings

Scale drawings use proportional relationships to represent actual dimensions.

Worked Example

A map uses a scale of:

1 cm : 5 km

Two towns are 7 cm apart on the map.

Find the actual distance.

Step 1: Set Up the Proportion

1/5 = 7/x

Step 2: Cross Multiply

x = 5 × 7

x = 35

Interpretation

The towns are 35 kilometres apart.

Important Observation

Always ensure that corresponding measurements occupy the same positions in the ratios.

Proportions in Shopping

Unit prices allow us to compare products and calculate costs.

Worked Example

A 3 kg bag of rice costs $9.

How much would 8 kg cost at the same price per kilogram?

Step 1: Write the Proportion

3/9 = 8/x

Step 2: Cross Multiply

3x = 9 × 8

3x = 72

Step 3: Solve

x = 24

Interpretation

Eight kilograms would cost $24 at the same unit price.

Checking with Unit Rate

9 ÷ 3 = $3/kg

8 × 3 = $24

Both methods agree.

Proportions in Speed and Distance

Proportions can be used when speed remains constant.

Worked Example

A cyclist travels 24 kilometres in 2 hours.

How far will the cyclist travel in 5 hours at the same average speed?

Step 1: Set Up the Proportion

24/2 = x/5

Step 2: Cross Multiply

24 × 5 = 2x

120 = 2x

Step 3: Solve

x = 60 km

Interpretation

The cyclist will travel 60 kilometres in 5 hours if the speed remains constant.

Important Observation

A proportion can only be used directly when the relevant relationship remains proportional.

If speed changes, the same proportion may no longer apply.

Proportional and Non-Proportional Situations

Not every relationship is proportional.

Example A: Proportional

A worker earns $15 per hour with no fixed bonus.

Earnings:

E = 15h

The ratio E/h is constant.

Therefore, the relationship is proportional.

Example B: Non-Proportional

A taxi charges a $5 starting fee plus $2 per kilometre.

Cost:

C = 2d + 5

The ratio C/d is not constant.

Therefore, the relationship is not proportional.

Comparison

Important Observation

A constant rate of change does not always mean that two quantities are proportional.

A proportional relationship must have the form y = kx.

Verifying Solutions to Proportions

After solving a proportion, substitute the answer into the original equation.

Example

Solve:

6/11 = 18/x

Cross multiply:

6x = 11 × 18

6x = 198

x = 33

Verify

6/11 = 18/33

Simplify 18/33:

18/33 = 6/11

The solution is correct.

Alternative Check

Cross products:

6 × 33 = 198

11 × 18 = 198

Both cross products are equal.

Important Observation

Checking solutions helps detect arithmetic mistakes, incorrect cross multiplication, and reversed ratios.

Choosing the Best Method

There are several ways to solve proportions.

Example

Solve:

4/7 = 12/x

Using equivalent ratios:

4 × 3 = 12

7 × 3 = 21

x = 21

Using cross multiplication:

4x = 7 × 12

4x = 84

x = 21

Both methods produce the same answer.

A General Method for Solving Proportions

Common Mistakes

Reversing one ratio

Corresponding quantities must appear in the same order in both ratios.

Cross multiplying incorrectly

Multiply each numerator by the opposite denominator.

Forgetting to solve the final equation

Cross multiplication produces an equation that must still be solved.

Ignoring units

Convert measurements when necessary before constructing proportions.

Assuming every linear relationship is proportional

A linear relationship with a nonzero starting value is not proportional.

Dividing only one term of a ratio

Equivalent ratios require the same multiplier or divisor for both terms.

Using a proportion when the relationship changes

Proportions require a constant multiplicative relationship.

Forgetting to verify

Substituting the answer helps confirm that the ratios are equivalent.

Confusing direct and inverse relationships

If one quantity doubles while another halves, the relationship may be inversely proportional rather than directly proportional.

Key Terms

• Proportion: Equation stating that two ratios are equal.

• Ratio: Comparison of quantities using division.

• Equivalent Ratios: Ratios representing the same comparison.

• Proportional Relationship: Relationship where one quantity is a constant multiple of another.

• Constant of Proportionality: Fixed ratio between corresponding quantities.

• Cross Multiplication: Method of solving proportions by multiplying diagonally across two fractions.

• Numerator: Top number of a fraction.

• Denominator: Bottom number of a fraction.

• Scale Factor: Multiplier used to produce an equivalent ratio or measurement.

• Unit Rate: Rate expressed per one unit of another quantity.

• Direct Proportion: Relationship where y = kx.

• Inverse Proportion: Relationship where the product of two quantities remains constant.

• Variable: Symbol representing an unknown or changing quantity.

• Equation: Mathematical statement showing equality.

• Verification: Process of checking whether a solution satisfies the original equation.

• Substitution: Replacing a variable with a known value.

• Scale Drawing: Drawing representing actual dimensions proportionally.

• Graph of a Proportion: Straight line passing through the origin.

• Cross Product: Product of a numerator and the opposite denominator in a proportion.

Key Takeaways

• A proportion states that two ratios are equal.

• Equivalent ratios can be created by multiplying or dividing both terms by the same nonzero number.

• Proportional relationships have a constant ratio between corresponding quantities.

• A proportional relationship can be written as y = kx.

• The graph of a proportional relationship is a straight line passing through the origin.

• Missing values can be calculated using equivalent ratios.

• Cross multiplication is useful when the scale factor is not obvious.

• The cross products of equal ratios are equal.

• Proportions are useful in recipes, shopping, maps, and constant-speed problems.

• Not every linear relationship is proportional.

• Units and corresponding quantities must be arranged consistently.

• Solutions should always be checked by substitution or equivalent ratios.

• Mathematical answers should be interpreted using the real-world context.

Check Your Understanding

Identifying Proportions

1. What is a proportion?

2. Are 2/3 and 6/9 equivalent ratios?

3. Is y = 4x a proportional relationship?

4. Is y = 4x + 2 a proportional relationship?

5. What must be true about y/x in a proportional relationship?

Equivalent Ratios

6. Complete 3 : 5 = 12 : x.

7. Complete 4 : 7 = x : 21.

8. Are 8 : 12 and 10 : 15 equivalent?

9. Find an equivalent ratio to 5 : 9 where the first term is 20.

10. Simplify 18 : 30.

Cross Multiplication

11. Solve 3/4 = x/20.

12. Solve 5/8 = 15/x.

13. Solve 7/9 = x/27.

14. Solve 2/5 = (x + 1)/15.

15. Solve 4/x = 12/21.

Practical Problems

16. A recipe uses 2 cups of flour for 8 muffins. How much flour is needed for 20 muffins?

17. A map uses a scale of 1 cm : 4 km. What actual distance does 6 cm represent?

18. Five notebooks cost $15. How much would 9 notebooks cost at the same unit price?

19. A cyclist travels 36 km in 3 hours. How far will the cyclist travel in 5 hours at the same speed?

20. Explain one method for verifying a proportion.

Answer Key

Extended Challenge

1. Solve 7/12 = x/36.

2. Solve 9/x = 15/25.

3. Solve (x + 3)/8 = 5/4.

4. Solve 2.5/6 = x/18.

5. Determine whether the table (2, 7), (4, 14), (6, 21), (8, 28) represents a proportional relationship.

6. A recipe requires 450 g of flour for 18 cookies. How much flour is required for 30 cookies?

7. A map uses a scale of 1 : 250,000. Find the actual distance represented by 6 cm.

8. A worker earns $135 for 9 hours. Calculate the earnings for 14 hours at the same hourly rate.

9. A machine produces 240 items in 8 minutes. How many items will it produce in 15 minutes at the same rate?

10. Explain why a taxi fare consisting of a fixed fee plus a distance charge is not directly proportional to distance.

Final Investigation: Planning a School Fundraiser

A school is organising a fundraiser and needs to purchase supplies, prepare food, and arrange transportation.

The organisers must use proportions to calculate quantities and costs.

Part A: Food Preparation

A recipe makes 12 servings using:

• 300 g flour

• 200 mL milk

• 4 eggs

The school needs 48 servings.

1. Determine the scale factor.

2. Calculate the amount of flour required.

3. Calculate the amount of milk required.

4. Calculate the number of eggs required.

5. Verify that all ingredient quantities have been scaled proportionally.

Part B: Purchasing Supplies

A supplier sells 5 notebooks for $12.50.

6. Calculate the unit price.

7. Write a proportion to determine the cost of 18 notebooks.

8. Solve the proportion using cross multiplication.

9. Verify the answer using the unit price.

10. Explain why the cost is proportional to the number of notebooks.

Part C: Transportation

A bus travels 120 kilometres in 2 hours at a constant average speed.

11. Calculate the average speed.

12. Write a proportion to determine the distance travelled in 3.5 hours.

13. Solve the proportion.

14. Calculate the time required to travel 300 kilometres at the same speed.

15. Explain why the proportion may not accurately predict travel time if traffic conditions change.

Part D: Scale Drawings

A school hall is represented on a plan using the scale:

1 cm : 2 m

The hall measures 9 cm by 6 cm on the plan.

16. Calculate the actual length.

17. Calculate the actual width.

18. Determine the actual floor area.

19. Explain why the scale factor must be applied to both dimensions.

20. Write a conclusion explaining how proportions help solve practical problems.

Investigation Check

Food preparation

• Scale factor = 4

• Flour = 1200 g

• Milk = 800 mL

• Eggs = 16

Supplies

• Unit price = $2.50

• Cost of 18 notebooks = $45

Transportation

• Average speed = 60 km/h

• Distance in 3.5 hours = 210 km

• Time for 300 km = 5 hours

Scale drawing

• Actual length = 18 m

• Actual width = 12 m

• Actual floor area = 216 m²