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?