What Is a Ratio?

A ratio compares two or more quantities.

Suppose a basket contains:

3 red apples and 5 green apples

The ratio of red apples to green apples is:

3 : 5

This tells us that for every 3 red apples, there are 5 green apples.

A ratio describes a relationship between quantities rather than simply describing one quantity by itself.

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The Order of a Ratio Matters

Consider:

3 red counters and 5 blue counters

The ratio of red to blue is:

3 : 5

But the ratio of blue to red is:

5 : 3

These ratios describe different comparisons.

Always read the question carefully to determine the required order.

red : blue = 3 : 5

blue : red = 5 : 3

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Ways to Write Ratios

Ratios can be written in several forms.

Suppose there are 2 cats for every 3 dogs.

The ratio can be written:

2 : 3

or:

2 to 3

It can also be represented using a fraction-like form:

2/3

However, when using fraction notation for a ratio, remember that it represents a comparison between the two quantities.

The colon form is often the clearest:

2 : 3

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Reading a Ratio

The ratio:

4 : 7

is read:

"four to seven."

It means that for every 4 units of the first quantity, there are 7 units of the second quantity.

For example:

boys : girls = 4 : 7

means:

For every 4 boys, there are 7 girls.

It does not necessarily mean there are exactly 4 boys and 7 girls.

There could be:

8 boys and 14 girls

12 boys and 21 girls

20 boys and 35 girls

All have the same ratio.

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Ratios Describe Relationships

Suppose a drink is made using:

1 part concentrate : 4 parts water

This means that for every 1 equal-sized part of concentrate, 4 equal-sized parts of water are used.

The actual size of a "part" can change.

If one part is 100 mL:

100 mL concentrate : 400 mL water

If one part is 250 mL:

250 mL concentrate : 1000 mL water

The relationship remains:

1 : 4

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Ratios Can Compare Part to Part

Suppose a bag contains:

4 red counters
6 blue counters

The ratio:

red : blue = 4 : 6

This compares one part of the group with another part.

This is called a part-to-part ratio.

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Ratios Can Compare Part to Whole

Using the same example:

4 red counters
6 blue counters

Total:

4 + 6 = 10 counters

The ratio of red counters to all counters is:

4 : 10

The ratio of blue counters to all counters is:

6 : 10

Be careful to identify what quantities are being compared.

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Ratio and Fraction: An Important Difference

Ratios and fractions are closely connected, but they are not always describing exactly the same relationship.

Suppose:

red : blue = 2 : 3

This means there are:

2 red parts

and:

3 blue parts.

Total parts:

2 + 3 = 5

Therefore, the fraction of the total that is red is:

2/5

not:

2/3

The ratio 2 : 3 compares red with blue.

The fraction 2/5 compares red with the total.

This distinction is very important.

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Representing Ratios with Objects

Ratios can be represented visually using objects.

Suppose we have:

2 circles and 3 squares.

Then:

circles : squares = 2 : 3

Visual models make the relationship easier to see.

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Representing Ratios with Bar Models

A ratio such as:

2 : 3

can be represented using equal-sized blocks.

First quantity:

□ □

Second quantity:

□ □ □

The first quantity contains 2 equal parts.

The second contains 3 equal parts.

Bar models are especially useful when solving ratio problems.

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Equivalent Ratios

Ratios can represent the same relationship even when the numbers are different.

For example:

2 : 3

is equivalent to:

4 : 6

because both parts were multiplied by 2.

Similarly:

2 : 3 = 6 : 9 = 8 : 12 = 10 : 15

These are called equivalent ratios.

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Creating Equivalent Ratios

To create an equivalent ratio, multiply or divide both parts by the same non-zero number.

For example:

3 : 5

Multiply both by 4:

3 × 4 : 5 × 4

Therefore:

12 : 20

So:

3 : 5 = 12 : 20

The quantities are larger, but their relationship remains unchanged.

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Simplifying Ratios

Ratios can often be simplified.

Consider:

12 : 18

Both numbers are divisible by 6.

Divide both parts by 6:

12 ÷ 6 : 18 ÷ 6

Therefore:

2 : 3

So:

12 : 18 = 2 : 3

The ratio 2 : 3 is in simplest form.

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Simplifying Using the Greatest Common Factor

Suppose:

24 : 36

The greatest common factor of 24 and 36 is:

12

Divide:

24 ÷ 12 = 2

36 ÷ 12 = 3

Therefore:

24 : 36 = 2 : 3

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Ratios with More Than Two Quantities

Ratios can compare more than two quantities.

For example, a paint mixture might use:

red : blue : white = 2 : 3 : 1

This means:

For every 2 parts red paint,

there are 3 parts blue paint,

and 1 part white paint.

The order remains important.

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Ratios in Tables

Ratio tables help organize equivalent ratios.

Suppose:

apples : oranges = 2 : 3

We can build equivalent pairs:

Each row represents the same ratio:

2 : 3

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How Ratio Tables Work

Starting with:

2 : 3

Multiply both quantities by 2:

4 : 6

Multiply both by 3:

6 : 9

Multiply both by 5:

10 : 15

Ratio tables are useful for recipes, prices, distances, rates, scale drawings, and many other situations.

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Finding a Missing Quantity

Suppose:

pens : pencils = 2 : 5

If there are 8 pens, how many pencils should there be to maintain the same ratio?

Start:

2 : 5

Since:

2 × 4 = 8

multiply the second quantity by 4 as well:

5 × 4 = 20

Therefore:

8 : 20

There should be:

20 pencils

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Ratios and Scaling

Ratios remain equivalent when both quantities are scaled by the same factor.

Suppose a recipe uses:

flour : sugar = 3 : 2

If the original recipe uses:

300 g flour

then each ratio part represents:

300 ÷ 3 = 100 g

Sugar:

2 × 100 = 200 g

Therefore:

300 g flour : 200 g sugar

which simplifies back to:

3 : 2

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Ratios in Recipes

Recipes frequently use ratios.

Suppose pancake batter uses:

flour : milk = 2 : 3

If we use 4 cups of flour:

2 → 4

The scale factor is:

× 2

Therefore:

3 × 2 = 6

We need:

6 cups of milk

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Ratios in Drinks and Mixtures

Suppose a drink is mixed using:

syrup : water = 1 : 5

If we use 3 cups of syrup:

Scale factor:

1 → 3 = ×3

Therefore:

5 × 3 = 15

Water needed:

15 cups

The mixture is:

3 : 15

which simplifies to:

1 : 5

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Ratios and Total Parts

Suppose:

boys : girls = 2 : 3

Total ratio parts:

2 + 3 = 5

This means:

Boys represent 2 of the 5 total parts.

Girls represent 3 of the 5 total parts.

Therefore:

Fraction boys:

2/5

Fraction girls:

3/5

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Finding Quantities from a Total

Suppose:

cats : dogs = 2 : 3

There are:

25 animals altogether

Total ratio parts:

2 + 3 = 5

Value of one part:

25 ÷ 5 = 5

Cats:

2 × 5 = 10

Dogs:

3 × 5 = 15

Check:

10 + 15 = 25

Therefore:

10 cats and 15 dogs

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Another Total-Quantity Example

A bag contains red and blue counters in the ratio:

3 : 7

There are:

50 counters altogether

Total parts:

3 + 7 = 10

One part:

50 ÷ 10 = 5

Red:

3 × 5 = 15

Blue:

7 × 5 = 35

Therefore:

15 red and 35 blue counters

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Comparing Quantities Using Ratios

Suppose Class A has:

12 computers for 24 students.

Class B has:

15 computers for 25 students.

Class A:

computers : students = 12 : 24

Simplify:

1 : 2

Class B:

15 : 25

Simplify:

3 : 5

These ratios can help compare access to computers.

Another useful method is to compare the number of computers per student.

Class A:

12/24 = 0.5

Class B:

15/25 = 0.6

The second class has a greater number of computers relative to its number of students.

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Comparing Ratios by Finding Equivalent Ratios

Compare:

2 : 3

and:

3 : 5

We can create equivalent ratios with the same first quantity.

For:

2 : 3

multiply by 3:

6 : 9

For:

3 : 5

multiply by 2:

6 : 10

Now compare:

6 : 9

and:

6 : 10

The relationships are easier to examine when one quantity is held constant.

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Ratios on Double Number Lines

A double number line can show how two quantities change together.

Suppose:

2 notebooks cost $6

Equivalent values include:

2 notebooks → $6

4 notebooks → $12

6 notebooks → $18

8 notebooks → $24

The relationship remains constant as both quantities increase proportionally.

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Ratios and Rates

A ratio compares quantities.

A rate is a ratio that compares quantities with different units.

For example:

150 km : 3 h

can be written as:

150 km / 3 h

Simplify:

50 km/h

This is a unit rate because it describes the quantity for one unit of another quantity.

Ratios provide the foundation for understanding rates.

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Ratio vs Rate

Consider:

boys : girls = 3 : 4

Both quantities count people.

This is a ratio.

Now consider:

distance : time = 120 km : 2 h

The units are different.

This creates a rate:

60 km/h

Many real-world proportional relationships involve rates.

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Ratios and Scale Drawings

Maps and diagrams use ratios to compare a drawing with the real object.

Suppose a map scale is:

1 cm : 5 km

This means:

1 cm on the map represents 5 km in reality.

If two places are 4 cm apart on the map:

4 × 5 km = 20 km

The actual distance is:

20 km

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Ratios in Models

Models also use ratios.

A model car might use a scale of:

1 : 20

This means:

1 unit on the model represents 20 of the same units on the actual car.

If a feature is:

8 cm

on the model, its real length is:

8 × 20 = 160 cm

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Ratios in Sports

Ratios can describe performance.

Suppose a basketball player makes:

8 successful shots from 12 attempts

Successes to attempts:

8 : 12

Simplify:

2 : 3

This can also be represented as a fraction:

8/12 = 2/3

or approximately:

66.7%

Different representations can communicate different aspects of the same information.

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Ratios in Science

Ratios appear throughout science.

Examples include:

• mass : volume
• distance : time
• reactants in chemical reactions
• genotype ratios
• scale drawings
• mechanical advantage
• concentration
• mixtures

For example:

density = mass/volume

is based on a ratio between mass and volume.

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Ratios in Maps and Scale

A map might use:

1 : 100,000

This means:

1 unit measured on the map represents 100,000 of the same units in reality.

For example:

1 cm on the map = 100,000 cm in reality

Since:

100,000 cm = 1 km

the scale can also be interpreted as:

1 cm : 1 km

Units must be handled carefully when interpreting scale ratios.

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Ratios in Probability

Suppose a bag contains:

3 red counters

and:

7 blue counters.

The ratio:

red : blue = 3 : 7

Total counters:

3 + 7 = 10

The probability of choosing red is:

3/10

Notice again:

3 : 7

compares red to blue.

But:

3/10

compares red with the total.

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Ratios and Percentages

Ratios can often be converted into percentages.

Suppose:

boys : girls = 3 : 2

Total parts:

3 + 2 = 5

Fraction that are boys:

3/5

Convert:

3/5 = 0.6 = 60%

Fraction that are girls:

2/5 = 0.4 = 40%

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Ratios with Different Units

Before simplifying some ratios, quantities need to be written using the same unit.

For example:

2 m : 50 cm

Do not immediately simplify 2 : 50.

First convert:

2 m = 200 cm

Therefore:

200 cm : 50 cm

Simplify:

4 : 1

So:

2 m : 50 cm = 4 : 1

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Another Unit Example

Simplify:

3 kg : 750 g

Convert:

3 kg = 3000 g

Now:

3000 : 750

Divide both by 750:

4 : 1

Therefore:

3 kg : 750 g = 4 : 1

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Ratios with Decimals

Ratios may contain decimals.

For example:

1.5 : 2

We usually prefer whole-number ratios.

Multiply both parts by 2:

3 : 4

Therefore:

1.5 : 2 = 3 : 4

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Ratios with Fractions

Ratios can also contain fractions.

For example:

1/2 : 3/4

Multiply both parts by 4:

2 : 3

Therefore:

1/2 : 3/4 = 2 : 3

The goal is often to rewrite the ratio using the simplest whole numbers.

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Explaining a Ratio in Context

Writing the ratio is only part of the task.

Suppose:

red : blue = 3 : 5

A good explanation is:

"For every 3 red counters, there are 5 blue counters."

A stronger explanation might add:

"This relationship remains the same for equivalent ratios such as 6 : 10 or 9 : 15."

Being able to explain the meaning of a ratio shows understanding beyond calculation.

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Worked Example 1

A box contains:

8 red pencils

12 blue pencils.

Write the ratio of red to blue in simplest form.

Start:

8 : 12

Divide both by 4:

2 : 3

Answer:

red : blue = 2 : 3

Meaning:

For every 2 red pencils, there are 3 blue pencils.

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Worked Example 2

A class has:

12 boys

18 girls.

Find the ratio of boys to girls.

12 : 18

Simplify by dividing by 6:

2 : 3

Therefore:

boys : girls = 2 : 3

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Worked Example 3

Using the previous class:

12 boys + 18 girls = 30 students.

Find the ratio of boys to all students.

12 : 30

Simplify:

2 : 5

Notice:

boys : girls = 2 : 3

but:

boys : total = 2 : 5

These are different comparisons.

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Worked Example 4

A recipe uses flour and sugar in the ratio:

5 : 2

If 15 cups of flour are used, how much sugar is needed?

Scale factor:

5 → 15 = ×3

Therefore:

2 × 3 = 6

Answer:

6 cups of sugar

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Worked Example 5

Red, yellow, and blue beads are in the ratio:

2 : 3 : 5

There are 40 beads altogether.

Total ratio parts:

2 + 3 + 5 = 10

One part:

40 ÷ 10 = 4

Red:

2 × 4 = 8

Yellow:

3 × 4 = 12

Blue:

5 × 4 = 20

Check:

8 + 12 + 20 = 40

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Worked Example 6

A map has scale:

1 cm : 8 km

Two towns are:

6.5 cm

apart on the map.

Actual distance:

6.5 × 8 = 52 km

Answer:

52 km

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Worked Example 7

Simplify:

1.2 : 1.8

Multiply both by 10:

12 : 18

Simplify:

2 : 3

Therefore:

1.2 : 1.8 = 2 : 3

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Worked Example 8: Explaining a Ratio

A paint mixture uses:

blue : white = 2 : 5

This means:

"For every 2 equal parts of blue paint, 5 equal parts of white paint are used."

If we doubled the mixture:

4 : 10

If we tripled it:

6 : 15

The color proportions remain unchanged because the ratios are equivalent.

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Checking Whether Ratios Are Equivalent

Are:

4 : 6

and:

10 : 15

equivalent?

Simplify the first:

4 : 6 = 2 : 3

Simplify the second:

10 : 15 = 2 : 3

Therefore:

4 : 6 = 10 : 15

The ratios are equivalent.

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Another Way to Check Equivalent Ratios

We can compare:

a : b

and:

c : d

by comparing the fractions:

a/b

and:

c/d

For example:

4 : 6

and:

10 : 15

Compare:

4/6 = 2/3

10/15 = 2/3

Therefore, they are equivalent.

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Ratios Do Not Tell You the Actual Size

Suppose two groups both have the ratio:

2 : 3

Group A might contain:

2 red and 3 blue.

Group B might contain:

200 red and 300 blue.

The relationship is the same, but the groups are very different in size.

A ratio tells us about relative quantities, not necessarily their actual amounts.

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Common Mistakes

Mistake 1: Reversing the order

If:

cats : dogs = 2 : 5

then:

dogs : cats = 5 : 2

not 2 : 5.

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Mistake 2: Changing only one side

Incorrect:

2 : 3 → 4 : 3

To create an equivalent ratio, both quantities must be multiplied or divided by the same number.

Correct:

2 : 3 → 4 : 6

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Mistake 3: Confusing part-to-part with part-to-whole

If:

red : blue = 2 : 3

then red is:

2/5 of the total

not:

2/3 of the total

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Mistake 4: Forgetting to simplify

8 : 12

can be simplified to:

2 : 3

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Mistake 5: Simplifying quantities with different units before converting

For:

2 m : 50 cm

first convert to the same units:

200 cm : 50 cm = 4 : 1

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Mistake 6: Thinking a ratio gives exact quantities

A ratio of:

2 : 3

does not necessarily mean there are exactly 2 and 3 objects.

It describes the relationship between the quantities.

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Mistake 7: Adding or subtracting the same number

Equivalent ratios are produced through multiplication or division, not by adding the same amount to both parts.

For example:

2 : 3

is equivalent to:

4 : 6

but not:

4 : 5

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Did You Know?

Ratios are one of the foundations of proportional reasoning.

Ratios are used in:

• recipes
• maps
• scale drawings
• construction
• photography
• sports statistics
• probability
• science
• medicine
• engineering
• mixtures
• rates
• finance

Ratios also connect directly to fractions, decimals, percentages, rates, and proportions.

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Key Terms

• Ratio: Comparison between two or more quantities.
• Ratio notation: Writing a ratio using a colon, such as 3 : 5.
• Equivalent ratios: Ratios representing the same relationship.
• Simplest form: Ratio whose terms have no common factor greater than 1.
• Part-to-part ratio: Comparison between two parts of a whole.
• Part-to-whole ratio: Comparison between one part and the total.
• Ratio table: Table showing equivalent values for two related quantities.
• Bar model: Diagram using equal-sized sections to represent a ratio.
• Double number line: Two aligned number lines showing how related quantities scale together.
• Scale factor: Number by which both parts of a ratio are multiplied or divided.
• Rate: Ratio comparing quantities with different units.
• Unit rate: Rate expressed for one unit of a quantity.
• Scale: Ratio comparing dimensions in a model or drawing with actual dimensions.
• Proportion: Statement that two ratios are equivalent.

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Quick Ratio Guide

To write a ratio:

identify the quantities → put them in the requested order

Example:

5 cats and 8 dogs:

cats : dogs = 5 : 8

To simplify a ratio:

divide both terms by the same common factor

Example:

15 : 25 = 3 : 5

To create an equivalent ratio:

multiply or divide both terms by the same non-zero number

Example:

3 : 4 = 6 : 8

To find quantities from a total:

add ratio parts → divide total by total parts → multiply by each ratio term

Example:

2 : 3 with total 25

Total parts:

5

One part:

25 ÷ 5 = 5

Quantities:

10 and 15

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Key Takeaways

• A ratio compares two or more quantities.
• Ratios describe relationships between quantities.
• Ratios can be written as a : b, "a to b," or sometimes using fraction notation.
• The order of a ratio matters.
• Ratios can compare part to part or part to whole.
• Part-to-part and part-to-whole comparisons should not be confused.
• Equivalent ratios represent the same relationship.
• To create equivalent ratios, multiply or divide all terms by the same non-zero number.
• Ratios can be simplified using common factors.
• Ratios can contain two, three, or more quantities.
• Diagrams, bar models, tables, and double number lines can represent ratios visually.
• Ratio tables help show how quantities scale together.
• The total number of ratio parts is found by adding the terms.
• Ratios can be used to divide a total into proportional quantities.
• Quantities should normally be converted to the same units before simplifying a ratio.
• Ratios can be converted into fractions and percentages when appropriate.
• Ratios are used in recipes, mixtures, maps, models, science, sports, and many other real-world situations.
• A ratio does not necessarily tell us the actual size of the quantities.
• The most important question when interpreting a ratio is:

What two quantities are being compared, and what does their relationship mean in this situation?

What Does It Mean to Simplify a Ratio?

To simplify a ratio means to write the ratio using the smallest whole numbers that represent the same relationship.

For example:

8 : 12

can be simplified to:

2 : 3

because both 8 and 12 can be divided by 4:

8 ÷ 4 : 12 ÷ 4

= 2 : 3

The quantities have changed, but the relationship between them has not changed.

Therefore:

8 : 12 = 2 : 3

These are equivalent ratios.

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Equivalent Ratios

Equivalent ratios are ratios that describe the same proportional relationship.

For example:

2 : 3

4 : 6

6 : 9

8 : 12

10 : 15

are all equivalent.

Each ratio can be created by multiplying both parts of:

2 : 3

by the same number.

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Multiplying to Create Equivalent Ratios

Start with:

3 : 5

Multiply both terms by 2:

3 × 2 : 5 × 2

= 6 : 10

Multiply both terms by 3:

3 × 3 : 5 × 3

= 9 : 15

Multiply both terms by 10:

3 × 10 : 5 × 10

= 30 : 50

Therefore:

3 : 5 = 6 : 10 = 9 : 15 = 30 : 50

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Dividing to Simplify Ratios

To simplify a ratio, we do the opposite.

We divide both terms by the same common factor.

For example:

12 : 20

Both numbers are divisible by 2:

12 ÷ 2 : 20 ÷ 2

= 6 : 10

We can simplify again:

6 ÷ 2 : 10 ÷ 2

= 3 : 5

Therefore:

12 : 20 = 3 : 5

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Why Must We Divide Both Terms?

A ratio describes a relationship.

If we change only one term, the relationship changes.

For example:

6 : 10

Divide both by 2:

3 : 5

This is equivalent.

But if we divide only the first term:

6 : 10 → 3 : 10

the relationship has changed.

Therefore:

6 : 10 ≠ 3 : 10

To maintain an equivalent ratio, multiply or divide every term by the same non-zero number.

---

Common Factors

A factor is a whole number that divides exactly into another whole number.

Consider:

12 and 18

Factors of 12:

1, 2, 3, 4, 6, 12

Factors of 18:

1, 2, 3, 6, 9, 18

Their common factors are:

1, 2, 3, 6

The greatest common factor is:

6

Therefore:

12 : 18

can be simplified by dividing both terms by 6:

12 ÷ 6 : 18 ÷ 6

= 2 : 3

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Greatest Common Factor

The greatest common factor (GCF) is the largest whole number that divides exactly into both numbers.

Using the GCF often lets us simplify a ratio in one step.

Example:

24 : 36

The GCF of 24 and 36 is:

12

Divide both terms:

24 ÷ 12 : 36 ÷ 12

= 2 : 3

Therefore:

24 : 36 = 2 : 3

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Simplifying in Several Steps

You do not have to identify the GCF immediately.

Consider:

48 : 72

Both numbers are even, so divide by 2:

24 : 36

Divide by 2 again:

12 : 18

Divide by 6:

2 : 3

Therefore:

48 : 72 = 2 : 3

As long as both terms are divided by the same factor, the ratio remains equivalent.

---

How Do We Know a Ratio Is Fully Simplified?

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

Consider:

6 : 15

Both are divisible by 3:

6 ÷ 3 : 15 ÷ 3

= 2 : 5

The factors of 2 are:

1, 2

The factors of 5 are:

1, 5

Their only common factor is 1.

Therefore:

2 : 5

is in simplest form.

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Visualizing Equivalent Ratios

Suppose we have:

4 red counters and 6 blue counters

The ratio is:

4 : 6

We can group them into two identical sets.

Each set contains:

2 red counters and 3 blue counters

Therefore:

4 : 6 = 2 : 3

The visual arrangement shows why the ratios are equivalent.

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Using Bar Models

Consider:

6 : 9

We can represent this using equal blocks.

First quantity:

6 equal units

Second quantity:

9 equal units

Since both numbers can be divided into groups of 3:

6 ÷ 3 = 2

9 ÷ 3 = 3

the simplified bar model contains:

2 parts : 3 parts

Therefore:

6 : 9 = 2 : 3

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Using Ratio Tables

Equivalent ratios can also be represented in a ratio table.

Suppose the simplest ratio is:

2 : 5

Every row represents the same relationship.

For example:

8 : 20

simplifies to:

2 : 5

because both terms can be divided by 4.

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Moving Up and Down a Ratio Table

A ratio table can be read in both directions.

Starting with:

3 : 4

multiply by 5:

15 : 20

To simplify:

15 : 20

divide both terms by 5:

3 : 4

Multiplication creates larger equivalent ratios.

Division can produce simpler equivalent ratios.

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Simplifying Ratios with Large Numbers

Consider:

120 : 180

Both numbers are divisible by 10:

12 : 18

Both are then divisible by 6:

2 : 3

Therefore:

120 : 180 = 2 : 3

Another method is to identify the GCF directly.

The GCF of 120 and 180 is:

60

Therefore:

120 ÷ 60 : 180 ÷ 60

= 2 : 3

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Simplifying Three-Part Ratios

Ratios can contain more than two terms.

For example:

12 : 18 : 24

Find a common factor of all three numbers.

All are divisible by 6:

12 ÷ 6 : 18 ÷ 6 : 24 ÷ 6

= 2 : 3 : 4

Therefore:

12 : 18 : 24 = 2 : 3 : 4

Every term must be divided by the same number.

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Checking Three-Part Ratios

Consider:

8 : 12 : 20

All three terms are divisible by 4:

2 : 3 : 5

Can this be simplified further?

The numbers 2, 3, and 5 have no common factor greater than 1.

Therefore:

2 : 3 : 5

is in simplest form.

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Ratios with Units

Sometimes ratios compare measurements.

Before simplifying, check the units.

For example:

2 m : 50 cm

We cannot simply simplify:

2 : 50

because the measurements use different units.

First convert to the same unit.

2 m = 200 cm

Therefore:

200 cm : 50 cm

Now divide both terms by 50:

4 : 1

Therefore:

2 m : 50 cm = 4 : 1

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Another Units Example

Simplify:

1.5 kg : 500 g

Convert:

1.5 kg = 1500 g

Now:

1500 : 500

Divide both by 500:

3 : 1

Therefore:

1.5 kg : 500 g = 3 : 1

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Simplifying Ratios with Decimals

Ratios containing decimals are usually easier to work with after converting them to whole numbers.

Consider:

1.5 : 2.5

Multiply both terms by 10:

15 : 25

Now divide both by 5:

3 : 5

Therefore:

1.5 : 2.5 = 3 : 5

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Another Decimal Example

Simplify:

0.8 : 1.2

Multiply both by 10:

8 : 12

Divide both by 4:

2 : 3

Therefore:

0.8 : 1.2 = 2 : 3

Multiplying both terms by 10 does not change their relationship.

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Simplifying Ratios with Fractions

Ratios may also contain fractions.

Consider:

1/2 : 3/4

One useful method is to multiply both terms by the lowest common denominator.

The denominators are 2 and 4.

The lowest common denominator is:

4

Multiply both terms by 4:

4 × 1/2 : 4 × 3/4

= 2 : 3

Therefore:

1/2 : 3/4 = 2 : 3

---

Another Fraction Ratio

Simplify:

2/3 : 5/6

The lowest common denominator is:

6

Multiply both terms by 6:

6 × 2/3 : 6 × 5/6

= 4 : 5

Therefore:

2/3 : 5/6 = 4 : 5

---

Equivalent Ratios in Recipes

Suppose a recipe uses:

flour : sugar = 6 : 4

Simplify:

Both terms are divisible by 2.

6 : 4 = 3 : 2

This means that for every:

3 parts flour

there are:

2 parts sugar

A larger recipe might use:

12 : 8

which also simplifies to:

3 : 2

---

Equivalent Ratios in Maps

Suppose a scale drawing uses:

5 cm : 20 m

The ratio can describe the same scale as:

1 cm : 4 m

because both quantities were divided by 5.

The simplified ratio can make the scale easier to interpret:

every 1 cm represents 4 m

When simplifying measurements, keep track of units carefully.

---

Equivalent Ratios in Groups

Suppose a club contains:

18 adults and 12 children

The ratio:

adults : children = 18 : 12

Divide both terms by 6:

3 : 2

Therefore:

18 : 12 = 3 : 2

This means:

For every 3 adults, there are 2 children in proportional terms.

It does not mean the club contains only 3 adults and 2 children.

---

Verifying Equivalent Ratios

After simplifying a ratio, we should be able to verify that the original and simplified ratios describe the same relationship.

There are several ways to check.

---

Method 1: Reverse the Scaling

Suppose:

15 : 25 = 3 : 5

To verify, ask:

What multiplies 3 to make 15?

3 × 5 = 15

Apply the same factor to the second term:

5 × 5 = 25

Therefore:

3 : 5 → 15 : 25

Both terms use the same scale factor.

The ratios are equivalent.

---

Method 2: Compare Fractions

Ratios:

8 : 12

and:

2 : 3

can be represented as:

8/12

and:

2/3

Simplify:

8/12 = 2/3

Therefore, the ratios are equivalent.

---

Method 3: Cross Products

For two-term ratios, we can check equivalence using cross products.

Compare:

4 : 6

and:

10 : 15

Write:

4/6 = 10/15

Calculate the cross products:

4 × 15 = 60

6 × 10 = 60

Since the cross products are equal:

4 : 6 = 10 : 15

---

Method 4: Compare Decimal Values

Consider:

3 : 4

and:

9 : 12

Calculate:

3 ÷ 4 = 0.75

and:

9 ÷ 12 = 0.75

Since the values are equal, the ratios are equivalent.

This method can be useful, although exact fraction or scaling methods are often preferable.

---

Checking Visually

Consider:

2 : 3

and:

6 : 9

The second ratio contains three groups of the first ratio:

2 × 3 = 6

3 × 3 = 9

The visual pattern confirms that the relationship is unchanged.

---

Worked Example 1

Simplify:

16 : 24

The GCF is:

8

Divide:

16 ÷ 8 : 24 ÷ 8

= 2 : 3

Answer:

2 : 3

---

Worked Example 2

Simplify:

35 : 49

Both numbers are divisible by 7:

35 ÷ 7 : 49 ÷ 7

= 5 : 7

Answer:

5 : 7

---

Worked Example 3

Simplify:

45 : 60

Both are divisible by 15:

45 ÷ 15 : 60 ÷ 15

= 3 : 4

Answer:

3 : 4

Check:

3 × 15 = 45

4 × 15 = 60

The relationship is preserved.

---

Worked Example 4

Simplify:

18 : 30 : 42

All terms are divisible by 6:

18 ÷ 6 : 30 ÷ 6 : 42 ÷ 6

= 3 : 5 : 7

Answer:

3 : 5 : 7

---

Worked Example 5

Simplify:

2.4 : 3.6

Multiply both by 10:

24 : 36

Divide by 12:

2 : 3

Therefore:

2.4 : 3.6 = 2 : 3

---

Worked Example 6

Simplify:

3/5 : 9/10

The lowest common denominator is 10.

Multiply both terms by 10:

6 : 9

Simplify:

2 : 3

Therefore:

3/5 : 9/10 = 2 : 3

---

Worked Example 7

Simplify:

3 m : 75 cm

Convert:

3 m = 300 cm

Therefore:

300 : 75

Divide both by 75:

4 : 1

Answer:

4 : 1

---

Worked Example 8: Verify the Result

A student says:

20 : 30 = 2 : 3

Check using scaling.

From:

2 : 3

multiply both terms by 10:

20 : 30

Correct.

Check using cross products:

20 × 3 = 60

30 × 2 = 60

Correct.

Therefore:

20 : 30 = 2 : 3

---

Spotting Non-Equivalent Ratios

Consider:

4 : 6

and:

8 : 10

Someone might think they are equivalent because both numbers increased.

But:

4 : 6 = 2 : 3

while:

8 : 10 = 4 : 5

These are different ratios.

Also:

4 × 10 = 40

but:

6 × 8 = 48

The cross products are not equal.

Therefore:

4 : 6 ≠ 8 : 10

---

Why Adding the Same Number Does Not Work

Start with:

2 : 3

Add 2 to both terms:

4 : 5

But:

2/3 ≠ 4/5

Therefore:

2 : 3 ≠ 4 : 5

Equivalent ratios are created through multiplicative relationships.

We multiply or divide every term by the same factor.

We do not create equivalent ratios by simply adding or subtracting the same number.

---

Ratio Families

You can think of equivalent ratios as belonging to the same ratio family.

For example:

3 : 4

belongs to the family:

3 : 4

6 : 8

9 : 12

12 : 16

15 : 20

30 : 40

All of these describe the same proportional relationship.

The simplest member is:

3 : 4

---

A Reliable Simplifying Strategy

When simplifying a ratio:

Step 1: Check the order of the quantities.

Step 2: Make sure comparable quantities use the same units.

Step 3: If necessary, remove decimals or fractions.

Step 4: Find a common factor.

Step 5: Divide every term by that factor.

Step 6: Repeat until there is no common factor greater than 1.

Step 7: Verify that the simplified ratio is equivalent to the original.

---

Common Mistakes

Mistake 1: Dividing only one term

Incorrect:

12 : 18 → 6 : 18

Correct:

12 : 18 → 6 : 9

Both terms must be divided by the same number.

---

Mistake 2: Stopping too early

For example:

12 : 18 → 6 : 9

This is equivalent, but it is not fully simplified.

Continue:

6 : 9 → 2 : 3

---

Mistake 3: Subtracting instead of dividing

Incorrect:

8 : 12 → 4 : 8

by subtracting 4 from both terms.

This changes the relationship.

Correct:

Divide both by 4:

8 : 12 → 2 : 3

---

Mistake 4: Ignoring units

Incorrect approach:

2 m : 50 cm → 2 : 50

Correct:

200 cm : 50 cm = 4 : 1

---

Mistake 5: Changing the order

If:

red : blue = 12 : 18

then the simplified ratio is:

2 : 3

not:

3 : 2

---

Mistake 6: Assuming smaller numbers automatically mean simplest form

Consider:

4 : 6

The numbers are small, but the ratio still simplifies:

4 : 6 = 2 : 3

---

Mistake 7: Simplifying different terms by different factors

Incorrect:

12 : 18

12 ÷ 4 : 18 ÷ 3 = 3 : 6

This changes the ratio.

The same factor must be applied to every term.

---

Did You Know?

Simplifying ratios is closely related to simplifying fractions.

For example:

12 : 18

can be associated with:

12/18

Simplifying:

12/18 = 2/3

Therefore:

12 : 18 = 2 : 3

Both processes use common factors to preserve the original proportional relationship.

This connection becomes important when studying:

• proportions
• percentages
• rates
• scale drawings
• probability
• similarity
• unit rates
• proportional equations

---

Key Terms

• Ratio: Comparison between two or more quantities.
• Equivalent ratios: Ratios representing the same proportional relationship.
• Simplify: Rewrite using smaller numbers while preserving the relationship.
• Simplest form: Ratio whose terms have no common factor greater than 1.
• Lowest terms: Another term for simplest form.
• Factor: Whole number that divides exactly into another whole number.
• Common factor: Factor shared by two or more numbers.
• Greatest common factor (GCF): Largest factor shared by all terms being simplified.
• Scale factor: Number used to multiply or divide every term of a ratio.
• Ratio table: Table displaying equivalent ratios.
• Bar model: Diagram showing quantities as proportional groups.
• Equivalent: Having the same mathematical value or relationship.
• Cross product: Product formed by multiplying diagonally across two equal ratios.
• Proportional relationship: Relationship in which quantities maintain a constant ratio.

---

Quick Guide

To simplify:

12 : 20

Find a common factor:

4

Divide:

12 ÷ 4 : 20 ÷ 4

Result:

3 : 5

To verify:

3 × 4 : 5 × 4

= 12 : 20

Therefore:

12 : 20 = 3 : 5

---

Key Takeaways

• Simplifying a ratio means expressing it using the smallest whole numbers that preserve the relationship.
• Equivalent ratios represent the same proportional relationship.
• To simplify a ratio, divide every term by the same common factor.
• Using the greatest common factor can simplify a ratio in one step.
• A ratio is fully simplified when its terms have no common factor greater than 1.
• You can also simplify gradually using several common factors.
• Equivalent ratios can be represented with objects, bar models, tables, and other diagrams.
• Multiplying every term by the same factor creates an equivalent ratio.
• Dividing every term by the same factor produces an equivalent ratio.
• Adding or subtracting the same number does not generally produce equivalent ratios.
• Three-part ratios can be simplified by dividing every term by the same common factor.
• Ratios containing different units should be converted to the same units before simplifying.
• Decimal ratios can often be converted to whole numbers before simplifying.
• Fraction ratios can often be simplified by multiplying by a common denominator.
• Simplified ratios can be verified using reverse scaling, fractions, cross products, or decimal comparisons.
• The order of the terms must remain unchanged.
• A simplified ratio does not change the underlying relationship.
• The central rule is:

Whatever you multiply or divide one term by, you must multiply or divide every term by the same non-zero number.

What Is a Rate?

A rate is a ratio that compares two quantities, usually with different units.

Examples include:

120 km in 2 hours

$15 for 3 kg

240 words in 4 minutes

600 mL in 5 seconds

Each rate tells us how two different quantities are related.

For example:

120 km / 2 h

compares distance with time.

This can be written as:

120 km : 2 h

or:

120 km / 2 h

---

Ratios and Rates

A ratio compares quantities.

A rate is a type of ratio that commonly compares quantities with different units.

For example:

boys : girls = 3 : 4

is a ratio.

But:

distance : time = 150 km : 3 h

is a rate because the quantities have different units.

The units are:

kilometres and hours

---

What Is a Unit Rate?

A unit rate is a rate in which the second quantity is 1 unit.

For example:

A car travels:

180 km in 3 hours

The rate is:

180 km / 3 h

Divide both quantities by 3:

60 km / 1 h

Therefore, the unit rate is:

60 km/h

The unit rate tells us how far the car travels in one hour.

---

Why Is It Called a Unit Rate?

The word unit means one.

A unit rate tells us the amount associated with one unit of another quantity.

Examples:

$3 per kilogram

70 km per hour

5 litres per minute

12 pages per hour

$2.50 per item

The word per is especially important.

Per means "for each" or "for every one."

---

Finding a Unit Rate

To find a unit rate, divide by the second quantity.

For example:

A cyclist travels:

72 km in 4 hours

Calculate:

72 km ÷ 4 h = 18 km/h

Therefore:

unit rate = 18 km/h

This means the cyclist travels an average of 18 kilometres for every hour.

---

Another Example

A pack of 6 notebooks costs:

$18

Find the cost per notebook.

Calculate:

$18 ÷ 6 = $3

Therefore:

unit rate = $3 per notebook

---

Units Matter

A rate should normally include its units.

For example:

60 km/h

means:

60 kilometres per hour

Writing only:

60

does not communicate the complete rate.

Other examples include:

$4/kg

20 L/min

5 m/s

80 words/min

$0.25/page

Units tell us what quantities are being compared.

---

Rates as Fractions

Rates can be written as fractions.

For example:

150 km in 3 hours

can be written:

150 km / 3 h

To find the unit rate:

150 ÷ 3 = 50

Therefore:

150 km / 3 h = 50 km / 1 h

or:

50 km/h

---

Rates in Tables

Rates can be represented using tables.

Suppose a machine produces 120 parts in 4 hours.

The unit rate is:

30 parts per hour

The table shows that the quantities increase proportionally.

---

Ratio Tables and Unit Rates

Suppose:

3 kg of apples cost $12

We can build a ratio table.

The unit rate is:

$4/kg

Once we know the unit rate, many other values are easy to calculate.

---

Rates on Double Number Lines

A double number line can represent two quantities that change together.

Suppose:

2 movie tickets cost $24

Then:

1 ticket → $12

2 tickets → $24

3 tickets → $36

4 tickets → $48

The unit rate is:

$12 per ticket

---

Unit Rates and Speed

Speed is one of the most common examples of a rate.

The basic equation is:

speed = distance / time

For example:

A car travels:

240 km in 4 hours

Calculate:

speed = 240 km / 4 h

speed = 60 km/h

Therefore:

60 km/h

is the average speed over that journey.

---

Understanding km/h

The unit:

km/h

means:

kilometres per hour

So:

80 km/h

means that at that rate, an object would travel 80 kilometres in one hour.

Similarly:

5 m/s

means:

5 metres for every 1 second.

---

Speed Example

A runner travels:

1500 m in 300 s

Calculate:

speed = 1500 / 300

= 5 m/s

Therefore:

average speed = 5 m/s

---

Finding Distance from a Rate

If we know the speed and time, we can calculate distance.

From:

speed = distance / time

we obtain:

distance = speed × time

Example:

A car travels at an average speed of:

70 km/h

for:

3 h

Distance:

70 × 3 = 210 km

Therefore:

distance = 210 km

---

Finding Time from a Rate

We can also rearrange:

speed = distance / time

to:

time = distance / speed

Example:

A cyclist travels:

60 km

at an average speed of:

20 km/h

Calculate:

time = 60 / 20

= 3 h

---

Cost as a Unit Rate

Unit rates are extremely useful when shopping.

Suppose:

Package A:

4 bottles for $10

Unit price:

$10 ÷ 4 = $2.50 per bottle

Package B:

6 bottles for $13.80

Unit price:

$13.80 ÷ 6 = $2.30 per bottle

The unit prices allow the packages to be compared on the same basis.

---

Why Total Price Alone Can Be Misleading

Suppose:

Small bag:

500 g for $4

Large bag:

800 g for $5.60

The large bag costs more overall, but that does not tell us which has the lower cost per gram.

Calculate cost per 100 g.

Small bag:

$4 ÷ 5 = $0.80 per 100 g

Large bag:

$5.60 ÷ 8 = $0.70 per 100 g

The larger bag has the lower unit price.

Unit rates make comparisons fairer when quantities differ.

---

Cost per Kilogram

Suppose 2.5 kg of rice costs:

$8.75

Find the cost per kilogram.

Calculate:

$8.75 ÷ 2.5 = $3.50

Therefore:

unit price = $3.50/kg

---

Cost per 100 g

Sometimes a different standard unit is more convenient.

If:

500 g costs $4

then:

100 g is one-fifth of 500 g.

Therefore:

$4 ÷ 5 = $0.80

So the rate is:

$0.80 per 100 g

Although this is not a unit rate in the strict "per 1 g" sense, it is a useful standardized rate for comparison.

---

Rates and Efficiency

Rates can also describe how efficiently something performs a task.

Suppose:

Machine A produces 240 parts in 6 hours.

Machine B produces 300 parts in 10 hours.

Machine A:

240 ÷ 6 = 40 parts/hour

Machine B:

300 ÷ 10 = 30 parts/hour

The unit rates provide a direct comparison of production rate.

In this specific measure, Machine A produces more parts per hour.

---

Efficiency Depends on What We Measure

The word efficiency can have different meanings.

One machine might produce more units per hour but use more electricity.

Another might produce fewer units per hour but use less energy.

Possible rates include:

products/hour

products/kWh

km/L

passengers/L

output/input

Therefore, always identify what measure of efficiency is being compared.

---

Fuel Economy

Suppose a vehicle travels:

600 km using 40 L of fuel

One useful rate is:

kilometres per litre

Calculate:

600 ÷ 40 = 15

Therefore:

15 km/L

This means the vehicle travels 15 km for each litre of fuel, on average under those conditions.

---

Litres per 100 Kilometres

Fuel use is also commonly expressed as:

L/100 km

Suppose a vehicle uses:

6 L/100 km

This means it uses 6 litres of fuel for every 100 kilometres travelled.

Notice that this rate behaves differently from km/L:

For km/L, a larger number means more distance per litre.

For L/100 km, a smaller number means less fuel used for the same distance.

Always understand what the units mean before comparing rates.

---

Rates in Work and Production

Suppose a printer produces:

150 pages in 5 minutes

Unit rate:

150 ÷ 5 = 30 pages/min

If the printer continues at the same rate for 8 minutes:

30 × 8 = 240 pages

Unit rates can be used to predict quantities when the rate remains constant.

---

Rates in Reading

A student reads:

84 pages in 3 hours

Average rate:

84 ÷ 3 = 28 pages/hour

At the same average rate, in 5 hours the student would read:

28 × 5 = 140 pages

This prediction assumes the reading rate remains approximately constant.

---

Rates in Typing

A person types:

450 words in 6 minutes

Unit rate:

450 ÷ 6 = 75 words/min

Therefore:

typing rate = 75 words per minute

---

Rates in Filling Containers

A pump transfers:

360 L in 6 minutes

Unit rate:

360 ÷ 6 = 60 L/min

At the same rate, in 10 minutes:

60 × 10 = 600 L

---

Rates in Science

Rates are important throughout science.

Examples include:

• speed: m/s
• acceleration: m/s²
• flow rate: L/min
• reaction rate: concentration/time
• population growth: organisms/year
• energy use: J/s
• power: W
• density: kg/m³ or g/cm³

A rate describes how one quantity relates to another.

---

Rates in Biology

A person's heart might beat:

72 beats/min

This means:

72 heartbeats for every minute.

A breathing rate might be:

15 breaths/min

These are unit rates because they describe the number of events per one unit of time.

---

Rates in Chemistry

Reaction rates describe how quickly reactants are consumed or products are formed.

For example, a simplified average rate might be expressed as:

0.20 mol/min

This means the measured amount changes by an average of 0.20 mol per minute over the specified interval.

Rates allow scientists to compare processes occurring over different time intervals.

---

Rates in Electricity

Electric power is another example of a rate.

1 watt = 1 joule per second

Therefore:

100 W

means energy is being transferred at a rate of:

100 J/s

This shows how rate concepts connect mathematics with physics.

---

Comparing Speeds Using Unit Rates

Cyclist A travels:

45 km in 3 h

Cyclist B travels:

64 km in 4 h

Cyclist A:

45 ÷ 3 = 15 km/h

Cyclist B:

64 ÷ 4 = 16 km/h

Converting both situations to kilometres per hour makes them directly comparable.

---

Comparing Costs Using Unit Rates

Store A:

8 pens for $12

Store B:

5 pens for $8

Store A:

$12 ÷ 8 = $1.50 per pen

Store B:

$8 ÷ 5 = $1.60 per pen

The unit rates reveal the cost for one pen.

---

Comparing Production Rates

Worker A produces:

84 items in 7 hours

Worker B produces:

100 items in 10 hours

Worker A:

84 ÷ 7 = 12 items/hour

Worker B:

100 ÷ 10 = 10 items/hour

The unit rate allows a direct comparison of production speed.

---

Comparing Different Time Units

Sometimes rates cannot be compared immediately because the units differ.

Suppose:

Runner A:

300 m/min

Runner B:

6 m/s

Convert Runner A to metres per second.

Since:

1 min = 60 s

Calculate:

300 ÷ 60 = 5 m/s

Now compare:

Runner A:

5 m/s

Runner B:

6 m/s

The rates can now be compared because they use the same units.

---

Units Must Match When Comparing Rates

Suppose one price is:

$4/kg

and another is:

$0.50 per 100 g

We should convert them to the same units.

Since:

1 kg = 1000 g = 10 × 100 g

the second price is:

10 × $0.50 = $5/kg

Now compare:

$4/kg

and:

$5/kg

Using the same units makes the comparison meaningful.

---

Unit Rates and Proportional Relationships

If a unit rate remains constant, the quantities form a proportional relationship.

Suppose:

1 kg → $3

Then:

2 kg → $6

3 kg → $9

4 kg → $12

The relationship can be written:

cost = 3 × mass

or:

C = 3m

The unit rate, 3, is the constant multiplier.

---

Unit Rates and Graphs

A constant unit rate can be represented by a straight line through the origin.

For example:

$4 per item

gives:

1 item → $4
2 items → $8
3 items → $12
4 items → $16

The constant rate determines how quickly the graph rises.

Later, this idea becomes important when studying slope and linear relationships.

---

Unit Rate as a Constant of Proportionality

Suppose:

5 notebooks cost $15

Unit rate:

15 ÷ 5 = 3

Therefore:

$3 per notebook

If:

x = number of notebooks

and:

y = cost

then:

y = 3x

The number:

3

is the constant of proportionality.

It tells us the cost for one notebook.

---

Unit Rate vs Average Rate

A calculated rate may describe an average rather than what happened at every moment.

Suppose a car travels:

180 km in 3 hours

Average speed:

60 km/h

This does not necessarily mean the car travelled exactly 60 km/h during every moment of the journey.

The car may have:

• accelerated
• slowed down
• stopped
• travelled at different speeds

The unit rate describes the overall average relationship.

---

Worked Example 1: Speed

A train travels:

420 km in 6 hours

Find the average speed.

420 ÷ 6 = 70

Therefore:

average speed = 70 km/h

---

Worked Example 2: Unit Price

A box of 12 markers costs:

$21.60

Find the cost per marker.

$21.60 ÷ 12 = $1.80

Therefore:

unit price = $1.80 per marker

---

Worked Example 3: Production Rate

A machine produces:

540 bottles in 9 minutes

Find the production rate.

540 ÷ 9 = 60

Therefore:

60 bottles/min

---

Worked Example 4: Distance

A cyclist travels at:

18 km/h

for:

2.5 h

Calculate:

distance = speed × time

distance = 18 × 2.5

distance = 45 km

---

Worked Example 5: Time

A vehicle travels:

240 km

at an average speed of:

80 km/h

Calculate:

time = distance / speed

time = 240 / 80

time = 3 h

---

Worked Example 6: Comparing Prices

Package A:

750 g for $6

Package B:

1 kg for $7.50

Convert Package A to cost per kilogram.

750 g = 0.75 kg.

Therefore:

$6 ÷ 0.75 = $8/kg

Package B:

$7.50/kg

Comparing the unit prices gives:

$8/kg versus $7.50/kg

---

Worked Example 7: Comparing Speeds

Runner A travels:

5 km in 25 minutes

Runner B travels:

4 km in 18 minutes

Runner A:

5 ÷ 25 = 0.20 km/min

Runner B:

4 ÷ 18 ≈ 0.222 km/min

Using the same units makes the comparison possible.

---

Worked Example 8: Efficiency

Machine A produces:

500 units using 25 kWh

Machine B produces:

420 units using 20 kWh

Machine A:

500 ÷ 25 = 20 units/kWh

Machine B:

420 ÷ 20 = 21 units/kWh

The rates describe output per unit of energy.

This provides a useful measure of energy productivity.

---

Worked Example 9: Multi-Step Rate Problem

A tap fills:

180 L in 4 minutes

Find the unit rate:

180 ÷ 4 = 45 L/min

How much water would flow in 7 minutes at the same rate?

45 × 7 = 315 L

Therefore:

315 L

---

Worked Example 10: Working Backwards

A machine produces:

35 items/min

How long will it take to produce:

420 items?

Use:

time = quantity / rate

Calculate:

420 ÷ 35 = 12

Therefore:

12 minutes

---

Checking Whether a Rate Is Reasonable

Estimation can help detect errors.

Suppose:

198 km in 4 hours

We can estimate:

200 ÷ 4 = 50 km/h

So the exact rate should be close to:

50 km/h

Calculate:

198 ÷ 4 = 49.5 km/h

This is reasonable.

If a calculator showed:

495 km/h

we should immediately suspect an error.

---

Checking Units

Units can also help check calculations.

Suppose:

distance = 200 km

and:

time = 4 h

Then:

distance/time = km/h

So the expected unit is:

km/h

If the answer is written as:

50 hours

the units reveal that something has gone wrong.

---

Why Unit Rates Are Useful

Unit rates convert different situations to a common basis.

Instead of comparing:

$18 for 6 items

with:

$25 for 10 items

we can compare:

$3/item

with:

$2.50/item

This makes the relationship much easier to interpret.

Unit rates answer useful questions such as:

• How much for one?
• How far in one hour?
• How much fuel for a standard distance?
• How many products per hour?
• How many words per minute?
• How much energy per product?
• How much water per minute?

---

Choosing the Direction of the Rate

A rate can sometimes be expressed in more than one direction.

Suppose:

300 km using 20 L

We could calculate:

300 ÷ 20 = 15 km/L

or:

20 ÷ 300 ≈ 0.0667 L/km

Both are mathematically valid, but they answer different questions.

15 km/L asks:

How far can we travel using one litre?

0.0667 L/km asks:

How much fuel is used for one kilometre?

The most useful form depends on the problem.

---

Common Mistakes

Mistake 1: Dividing in the wrong order

To find dollars per item:

total cost ÷ number of items

not:

items ÷ dollars

Ask yourself what unit you want in the answer.

---

Mistake 2: Forgetting units

60

is incomplete if the answer should be:

60 km/h

---

Mistake 3: Comparing totals instead of unit rates

A larger package may cost more overall while having a lower cost per unit.

---

Mistake 4: Comparing rates with different units

Do not directly compare:

300 m/min

with:

6 m/s

Convert them to matching units first.

---

Mistake 5: Assuming a unit rate is always exact at every moment

A speed calculated from total distance and total time is usually an average speed.

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Mistake 6: Assuming a larger rate always means "better"

For:

km/L

larger generally means more distance per litre.

For:

L/100 km

smaller generally means less fuel used for the same distance.

Interpret the units.

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Mistake 7: Ignoring context

A production rate of 50 products/hour may look impressive, but quality, energy use, waste, and cost might also matter.

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Did You Know?

Rates connect many different areas of mathematics and science.

Examples include:

Speed

km/h or m/s

Flow rate

L/min

Typing speed

words/min

Heart rate

beats/min

Unit price

$/kg

Fuel economy

km/L

Power

J/s

Density

g/cm³

Understanding rates is also an important foundation for:

• proportions
• linear relationships
• slope
• dimensional analysis
• scientific formulas
• financial mathematics
• data analysis

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Key Terms

• Rate: Ratio comparing two quantities, usually with different units.
• Unit rate: Rate expressed for one unit of another quantity.
• Per: For each or for every one.
• Unit price: Cost per one item or standard unit of quantity.
• Speed: Distance travelled per unit of time.
• Average speed: Total distance divided by total time.
• Flow rate: Quantity of fluid transferred per unit of time.
• Production rate: Number of products produced per unit of time.
• Fuel economy: Rate comparing distance travelled and fuel used.
• Efficiency measure: Rate comparing useful output with a relevant input.
• Proportional relationship: Relationship in which two quantities maintain a constant ratio.
• Constant of proportionality: Constant unit rate connecting two proportional quantities.
• Ratio table: Table showing related proportional quantities.
• Double number line: Diagram showing how two related quantities scale together.

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Key Equations

Unit rate:

unit rate = quantity / number of units

Average speed:

speed = distance / time

Distance:

distance = speed × time

Time:

time = distance / speed

Unit price:

unit price = total cost / quantity

Production rate:

production rate = number produced / time

Flow rate:

flow rate = volume / time

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Unit Rate Problem-Solving Strategy

When solving a rate problem:

Step 1: Identify the two quantities being compared.

Step 2: Identify their units.

Step 3: Decide which unit rate is needed.

Step 4: Divide in the correct order.

Step 5: Include the units in the answer.

Step 6: If comparing rates, convert them to the same units.

Step 7: Estimate the expected value.

Step 8: Calculate.

Step 9: Check whether the answer is reasonable.

Step 10: Explain what the unit rate means in context.

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Key Takeaways

• A rate compares two quantities, usually with different units.
• A unit rate compares a quantity with exactly one unit of another quantity.
• The word per is commonly used to describe rates.
• Unit rates are found by division.
• Units are an essential part of a rate.
• Speed is a rate comparing distance with time.
• Unit price compares cost with quantity.
• Production rate compares output with time.
• Efficiency can be analyzed using rates such as output per unit of energy or fuel.
• Unit rates allow situations involving different quantities to be compared on a common basis.
• Rates should normally be converted to matching units before comparison.
• The direction of a rate matters: km/L and L/km answer different questions.
• A larger numerical rate does not automatically mean better; the meaning depends on the units and context.
• A constant unit rate represents a proportional relationship.
• In a proportional equation such as y = kx, the unit rate can act as the constant of proportionality k.
• Unit rates can be represented using tables, double number lines, equations, and graphs.
• Estimation helps determine whether a calculated rate is reasonable.
• A useful question when interpreting any rate is:

How much of one quantity is there for each one unit of the other quantity?

From the graph:

Sports : Music

30 : 20

Simplify:

3 : 2

Gaming : Art

25 : 15

Simplify:

5 : 3

Graphs provide the quantities, while ratio reasoning allows us to compare them.

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Interpreting Ratios from Graphs

When using a graph:

1. Identify the quantities being compared.

2. Read their values carefully.

3. Write the ratio in the requested order.

4. Simplify if appropriate.

5. Explain what the ratio means in context.

For example:

If a graph shows:

40 students prefer A

and:

24 students prefer B

then:

A : B = 40 : 24

Simplify:

5 : 3

Meaning:

For every 5 students preferring A, there are proportionally 3 preferring B.

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Ratios in Pie Charts

A pie chart represents parts of a whole.

Suppose a survey gives:

Walking = 40%

Bus = 30%

Car = 20%

Cycling = 10%

The ratio:

Walking : Bus = 40 : 30

Simplify:

4 : 3

The ratio:

Car : Cycling = 20 : 10

Simplify:

2 : 1

Percentages can therefore be converted into ratios for comparison.

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Ratios and Population Data

Suppose a region has:

60,000 adults

and:

20,000 children

The ratio:

adults : children = 60,000 : 20,000

Divide by 20,000:

3 : 1

This communicates the relative sizes of the two groups without needing to repeat the large numbers.

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Ratios in Science

Ratios are extremely important in science.

Examples include:

• mass : volume
• distance : time
• force : area
• reactants in chemical equations
• genetic ratios
• scale models
• concentrations
• energy input : useful output

For example:

A material with:

mass = 200 g

and:

volume = 50 cm³

has:

mass : volume = 200 : 50

Simplify:

4 : 1

Its density is:

4 g/cm³

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Ratios in Chemistry

Chemical equations contain ratios between reacting particles.

For example:

2H₂ + O₂ → 2H₂O

The particle ratio is:

H₂ : O₂ : H₂O = 2 : 1 : 2

This means the balanced equation represents:

2 hydrogen molecules reacting with 1 oxygen molecule to produce 2 water molecules.

Ratios become especially important later when studying stoichiometry.

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Ratios in Biology

Genetics frequently involves ratios.

For example, a simple genetic model might predict a phenotype ratio of:

3 : 1

This means that over many expected outcomes, approximately three parts may show one phenotype for every one part showing another, under the assumptions of that model.

It does not guarantee that every group of four offspring will contain exactly three of one type and one of another.

---

Ratios in Engineering

Engineers use ratios when designing:

• buildings
• bridges
• machines
• gears
• models
• technical drawings
• structures

A gear system might contain:

Gear A = 20 teeth

Gear B = 60 teeth

Ratio:

20 : 60

Simplify:

1 : 3

Gear ratios help engineers analyze how rotational speed and torque change between connected gears.

---

Ratios in Photography and Screens

Screen dimensions are often described using aspect ratios.

Common examples include:

16 : 9

and:

4 : 3

A 16 : 9 screen means that for every:

16 units of width

there are:

9 units of height

The actual screen could be small or large. The ratio describes its shape.

---

Ratios in Construction

Suppose concrete is mixed using:

cement : sand : aggregate = 1 : 2 : 4

If one ratio part represents:

5 kg

then:

Cement:

1 × 5 = 5 kg

Sand:

2 × 5 = 10 kg

Aggregate:

4 × 5 = 20 kg

Total:

35 kg

The ratio ensures that the components maintain the intended proportions.

---

Ratios in Probability

Suppose a bag contains:

6 red counters

and:

4 blue counters

Ratio:

red : blue = 6 : 4 = 3 : 2

Total counters:

10

Probability of red:

6/10 = 3/5

Probability of blue:

4/10 = 2/5

Remember:

3 : 2

compares red with blue.

But:

3/5

compares red with the total.

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Ratio vs Actual Quantity

Suppose two classrooms both have:

boys : girls = 2 : 3

Class A could contain:

10 boys and 15 girls.

Class B could contain:

20 boys and 30 girls.

The ratio is the same, but the actual numbers are different.

Ratios describe relative quantities, not necessarily group size.

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Comparing Different-Sized Groups

Suppose:

School A:

300 students, 30 computers.

School B:

500 students, 40 computers.

Simply comparing computers:

30 vs 40

suggests School B has more.

But compare computers to students.

School A:

30 : 300 = 1 : 10

School B:

40 : 500 = 2 : 25

This is equivalent to:

0.08 computers per student

compared with:

0.10 computers per student

Ratio and rate comparisons reveal information that totals alone may hide.

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Choosing the Useful Form

Different situations may be easier to understand using different representations.

For example:

3 : 5

might be useful when mixing paint.

3/8

might be useful when describing the fraction of the total represented by the first quantity.

37.5%

might be useful when reporting the first quantity as a percentage of the total.

0.6

might be useful when comparing the first quantity directly with the second through division.

The best representation depends on the question being asked.

---

Worked Example 1: Recipe

A recipe uses:

flour : sugar = 5 : 2

If 20 cups of flour are used:

Scale factor:

5 → 20 = ×4

Sugar:

2 × 4 = 8 cups

Answer:

8 cups of sugar

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Worked Example 2: Mixture

A solution uses:

concentrate : water = 1 : 6

You need:

21 L

altogether.

Total parts:

1 + 6 = 7

One part:

21 ÷ 7 = 3 L

Concentrate:

3 L

Water:

18 L

Check:

3 : 18 = 1 : 6

---

Worked Example 3: Map

Scale:

1 cm : 12 km

Map distance:

7.5 cm

Actual distance:

7.5 × 12 = 90 km

Answer:

90 km

---

Worked Example 4: Scale Drawing

A building is:

30 m high

A drawing uses:

1 cm : 5 m

Drawing height:

30 ÷ 5 = 6 cm

Answer:

6 cm

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Worked Example 5: Sports

Player A scores:

24 goals in 32 games

Player B scores:

30 goals in 50 games

Player A:

24/32 = 0.75 goals/game

Player B:

30/50 = 0.60 goals/game

Using rates allows their scoring records to be compared relative to games played.

---

Worked Example 6: Table

A table shows:

Calculate:

140 ÷ 2 = 70 km/h

280 ÷ 4 = 70 km/h

420 ÷ 6 = 70 km/h

The ratio is constant.

Therefore, the data represent a proportional relationship with an average rate of:

70 km/h

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Worked Example 7: Three-Part Ratio

A school club has students in the ratio:

Grade 8 : Grade 9 : Grade 10 = 2 : 3 : 4

There are:

45 students

altogether.

Total parts:

2 + 3 + 4 = 9

One part:

45 ÷ 9 = 5

Grade 8:

2 × 5 = 10

Grade 9:

3 × 5 = 15

Grade 10:

4 × 5 = 20

Check:

10 + 15 + 20 = 45

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Worked Example 8: Scale Model

A model bridge uses a scale:

1 : 200

A section measures:

35 cm

on the model.

Actual length:

35 × 200 = 7000 cm

Convert:

7000 cm = 70 m

Answer:

70 m

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Worked Example 9: Interpreting Data

A survey records:

Music = 48 students

Sports = 72 students

Art = 36 students

Find:

Sports : Music

Start:

72 : 48

Divide by 24:

3 : 2

Interpretation:

For every 3 students choosing sports, there are proportionally 2 students choosing music.

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Worked Example 10: Comparison

Drink A uses:

2 parts concentrate : 7 parts water

Drink B uses:

3 parts concentrate : 9 parts water

Compare the amount of concentrate relative to water.

Drink A:

2/7 ≈ 0.286

Drink B:

3/9 = 1/3 ≈ 0.333

Drink B has a greater amount of concentrate relative to its water content.

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A Reliable Real-World Ratio Strategy

When solving a practical ratio problem:

Step 1: Identify the quantities being compared.

Step 2: Write them in the correct order.

Step 3: Check the units.

Step 4: Convert units if necessary.

Step 5: Simplify the ratio if useful.

Step 6: Identify the scale factor or unit rate.

Step 7: Calculate the missing quantity.

Step 8: Include appropriate units.

Step 9: Check that the final quantities preserve the original ratio.

Step 10: Explain what the answer means in context.

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Checking Your Answer

Suppose:

red paint : white paint = 2 : 5

You calculate that:

8 L red requires 20 L white

Check:

8 : 20

Divide both by 4:

2 : 5

The original ratio has been preserved.

Therefore, the answer is consistent.

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Estimating Before Calculating

Suppose a map scale is:

1 cm : 9 km

and a route measures:

5.2 cm

Estimate:

5 × 9 ≈ 45 km

Exact calculation:

5.2 × 9 = 46.8 km

The exact answer is close to our estimate, so it appears reasonable.

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When Ratio Reasoning Does Not Apply

Not every real-world relationship is proportional.

Suppose a delivery company charges:

$5 fixed fee + $2 per kilometre

For 1 km:

$7

For 2 km:

$9

For 3 km:

$11

The ratios:

7/1, 9/2, 11/3

are not constant.

Therefore, total cost and distance are not directly proportional.

The fixed $5 charge changes the relationship.

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Common Mistakes

Mistake 1: Reversing the ratio

If the question asks for:

juice : water

do not write:

water : juice

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Mistake 2: Changing only one quantity

If a recipe doubles, every ingredient must be scaled appropriately.

---

Mistake 3: Forgetting total parts

For:

2 : 3

the total is:

5 parts

not 3 parts.

---

Mistake 4: Ignoring units

For:

1 cm : 2 m

the units are different and must be handled carefully during calculations.

---

Mistake 5: Comparing totals instead of relative values

40 successes may sound better than 30 successes, but the number of attempts also matters.

---

Mistake 6: Assuming every table is proportional

Check whether the ratio or unit rate remains constant.

---

Mistake 7: Assuming a ratio tells you the total

A ratio of:

3 : 2

could describe:

3 and 2

30 and 20

300 and 200

or infinitely many other equivalent pairs.

---

Mistake 8: Confusing part-to-part with part-to-whole

If:

red : blue = 3 : 2

then the fraction that is red is:

3/5

not:

3/2

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Why Ratios Are Powerful

Ratios allow us to take a known relationship and apply it to a different-sized situation.

A recipe can be doubled.

A map can represent an entire country on one page.

A model can represent a huge building.

Athletes can be compared despite playing different numbers of games.

Mixtures can be reproduced consistently.

Data from groups of different sizes can be compared.

The numbers change, but the relationship remains consistent.

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Did You Know?

Many objects around us depend on ratios.

A widescreen display may use an aspect ratio of:

16 : 9

A map may use a scale such as:

1 : 50,000

A model may use:

1 : 100

A recipe may use:

2 : 3

A sports statistic may compare:

successes : attempts

Ratios provide a common mathematical language for describing all of these relationships.

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Key Terms

• Ratio: Comparison between two or more quantities.
• Equivalent ratios: Ratios representing the same proportional relationship.
• Scale factor: Number used to multiply or divide related quantities.
• Map scale: Ratio comparing map distance with actual distance.
• Scale drawing: Drawing whose dimensions are proportional to the real object.
• Scale model: Smaller or larger representation of an object that preserves proportional dimensions.
• Mixture: Combination of two or more substances.
• Part-to-part ratio: Comparison between different parts of a group.
• Part-to-whole ratio: Comparison between one part and the total.
• Rate: Ratio comparing quantities, often with different units.
• Unit rate: Rate expressed per one unit.
• Proportional relationship: Relationship that maintains a constant ratio.
• Statistics: Numerical information collected and analyzed to describe data.
• Aspect ratio: Ratio comparing width with height.
• Concentration: Amount of one component relative to a mixture or solution.

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Real-World Ratio Guide

Recipes

Keep ingredient ratios constant when scaling.

Mixtures

Maintain the correct proportion of each component.

Maps

Use the scale to convert between map and actual distance.

Scale Drawings

Multiply or divide dimensions using the scale factor.

Sports

Compare performance relative to attempts, games, time, or opportunities.

Statistics

Use ratios to compare groups of different sizes.

Tables

Look for a constant ratio or unit rate.

Graphs

Read the quantities first, then form and simplify the required ratio.

Science

Use ratios to describe quantities such as mass-to-volume, distance-to-time, chemical proportions, and concentrations.

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Key Takeaways

• Ratios are used throughout everyday life to compare related quantities.
• Recipes use ratios to maintain consistent ingredient proportions.
• Mixture ratios allow substances to be combined consistently.
• Map scales compare distances on a map with actual distances.
• Scale drawings and models preserve proportional dimensions.
• Scale factors allow objects and relationships to be enlarged or reduced.
• Sports ratios allow performance to be compared relative to opportunities.
• Statistical ratios help compare groups of different sizes.
• Tables can show whether ratios remain constant.
• Graphs can provide quantities that can be compared using ratios.
• Ratios can often be converted into fractions, decimals, percentages, or unit rates.
• Units must be considered carefully when using maps and scale drawings.
• Part-to-part ratios should not be confused with part-to-whole fractions.
• Ratios describe relative quantities rather than necessarily describing the actual size of a group.
• Not every real-world relationship is proportional.
• A fixed fee or changing rate can prevent a relationship from maintaining a constant ratio.
• Answers should be checked by simplifying the final quantities back to the original ratio.
• Ratios help us solve real-world problems because they allow a known relationship to be scaled, compared, interpreted, and applied to new situations.