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.

---

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.

---

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

---

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

---

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

---

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

---

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.

---

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.

---

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

---

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.

---

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.

---

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

---

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.

---

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.

---

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

---

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

---

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

---

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.

---

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.