Ratios: Definition, Formula & Example

Ratios compare quantities through division and show their relative sizes.

A two-term ratio can be written:

a:b

a to b

or:

a/b

provided the second quantity is nonzero when the ratio is interpreted as division.

For example, if a group contains:

8 red objects

and:

12 blue objects

the red-to-blue ratio is:

8:12

Simplify both terms by:

4

to get:

2:3

This means that for every 2 units of the first quantity, there are 3 corresponding units of the second quantity.

Ratios can compare parts, wholes, rates, measurements, scaled quantities, mixtures, probabilities, sequence terms, and many other numerical relationships.

What Are Ratios?

A ratio compares quantities multiplicatively rather than merely subtracting them.

Suppose:

A = 6

B = 9

The ratio:

A:B = 6:9

simplifies:

2:3

This says:

A/B = 2/3

It does not say the difference is 3, even though:

9 – 6 = 3

Difference and ratio measure different kinds of relationships.

Basic Ratio Formula

For two quantities:

A and B

the ratio of A to B is:

A:B = A/B

For example:

15:20

corresponds to:

15/20

Simplify:

15/20 = 3/4

Therefore:

15:20 = 3:4

The singular ratio page focuses on this basic one-comparison formula. Here the broader emphasis is on different ratio structures, equivalent ratios, multi-part comparisons, scaling, rates, and applications.

Order Matters

The ratio:

2:5

is not the same as:

5:2

Numerically:

2/5 = 0.4

while:

5/2 = 2.5

For a group with:

2 cats

and:

5 dogs

cats to dogs:

2:5

dogs to cats:

5:2

Always follow the order stated in the problem.

Simplifying Ratios

Whole-number ratios are simplified by dividing every term by their greatest common factor.

Simplify:

30:45

Find:

GCF(30,45) = 15

Divide:

30/15 = 2

45/15 = 3

Therefore:

30:45 = 2:3

Equivalent Ratios

Equivalent ratios represent the same numerical comparison.

Starting from:

3:7

multiply both terms by 2:

6:14

Multiply both by 5:

15:35

Thus:

3:7 = 6:14 = 15:35

In general:

a:b = ka:kb

for nonzero scale factor k.

These equal ratio relationships naturally form proportions.

Ratio Tables

Equivalent ratios can be organized in a table.

Starting with:

2:5

Each row has:

First/Second = 2/5

Therefore every row represents the same ratio.

Find a Missing Ratio Value

Suppose:

4:7 = 20:x

The first term has been multiplied by:

5

Therefore multiply the second term by 5:

7 × 5 = 35

So:

x = 35

Alternatively, write the proportion:

4/7 = 20/x

Cross multiply:

4x = 140

x = 35

Part-to-Part Ratios

A part-to-part ratio compares two categories within the same overall group.

Suppose there are:

10 red balls

and:

15 blue balls

Red to blue:

10:15

Simplify:

2:3

This ratio compares the two parts directly.

Part-to-Whole Ratios

Using the same group:

Red = 10

Blue = 15

Total:

25

Red to total:

10:25

Simplify:

2:5

Blue to total:

15:25

Simplify:

3:5

Part-to-part:

2:3

Part-to-whole:

2:5

These answer different questions.

Convert Part-to-Part Ratio to Fractions of a Whole

Suppose two categories have ratio:

4:7

Total ratio parts:

4 + 7 = 11

Therefore the first category represents:

4/11

of the total.

The second represents:

7/11

of the total.

As percentages:

4/11 × 100% ≈ 36.36%

7/11 × 100% ≈ 63.64%

The percentages sum to:

100%

Three-Part Ratios

Ratios can compare three or more quantities.

For example:

2:3:5

means the quantities are proportional to:

2 parts

3 parts

5 parts

Total ratio units:

2 + 3 + 5 = 10

Therefore the three quantities represent:

20%

30%

50%

of the combined total.

Divide a Total in a Three-Part Ratio

Divide:

600

in the ratio:

2:3:5

Total ratio parts:

10

One part:

600/10 = 60

First share:

2 × 60 = 120

Second:

3 × 60 = 180

Third:

5 × 60 = 300

Therefore:

120:180:300

Check:

120 + 180 + 300 = 600

and dividing each by 60 recovers:

2:3:5

Ratios With Four or More Terms

The same method extends to:

a:b:c:d

For example:

1:2:3:4

has total parts:

10

If the total quantity is:

500

one part equals:

50

Therefore quantities are:

50, 100, 150, 200

These preserve the original four-term ratio.

Ratios and Rational Numbers

A two-term integer ratio:

a:b

with nonzero b has numerical value:

a/b

which is a rational number.

For example:

7:8

has value:

7/8

= 0.875

The contextual ratio compares two quantities, while the rational number is the numerical quotient of that comparison.

Ratios Within the Real Numbers

Ratio values that can be expressed as:

a/b

for integers a and nonzero b are rational and therefore belong to the real numbers.

But ratios can also compare real-valued measurements.

For example:

√2 : 1

has quotient:

√2

which is irrational but still real.

Thus the concept of a ratio is broader than ratios formed only from integer terms.

Ratios With Decimal Values

Simplify:

1.5:2.25

Multiply each term by 100:

150:225

Divide by:

75

giving:

2:3

You could also multiply first by only 4:

1.5 × 4 = 6

2.25 × 4 = 9

Then:

6:9 = 2:3

Any common nonzero scaling preserves the ratio.

Ratios With Fractions

Simplify:

2/3 : 5/6

Multiply both terms by the least common denominator:

6

Then:

2/3 × 6 = 4

5/6 × 6 = 5

Therefore:

2/3 : 5/6 = 4:5

This converts a fractional ratio into an equivalent whole-number ratio.

Ratios With Mixed Numbers

Simplify:

1 1/2 : 2 1/4

Convert:

1 1/2 = 3/2

2 1/4 = 9/4

Multiply both by 4:

6:9

Simplify:

2:3

The ratio has not changed; only its representation has.

Ratios With Units

Ratios should use compatible units when the quantities measure the same dimension.

Suppose:

2 meters : 50 centimeters

Convert:

2 meters = 200 centimeters

Then:

200:50

Simplify:

4:1

Without converting units first, writing:

2:50

would compare inconsistent measurement scales.

Ratios With Different Units

When quantities have different units, the comparison is usually called a rate.

For example:

180 km : 3 hours

gives:

60 km : 1 hour

or:

60 km/h

The mathematical structure is still division:

180/3 = 60

but the units remain meaningful.

Unit Ratios

A ratio with second term 1 can be useful for comparisons.

For:

24:6

divide both terms by 6:

4:1

For quantities with units:

$15 : 3 kg

becomes:

$5 : 1 kg

The unit form makes comparisons between alternatives easier.

Compare Two Unit Prices

Product A:

$12 for 4 units

Unit price:

12/4 = $3 per unit

Product B:

$20 for 5 units

Unit price:

20/5 = $4 per unit

Therefore Product A has the lower price-to-unit ratio.

The ratio calculation provides a common reference of one unit.

Comparing Ratios by Cross Multiplication

Compare:

5:8

and:

7:10

Write:

5/8

and:

7/10

Cross multiply:

5 × 10 = 50

8 × 7 = 56

Since:

50 < 56

we have:

5/8 < 7/10

Therefore:

5:8 < 7:10

Comparing Ratios Using Decimals

The same ratios give:

5/8 = 0.625

7/10 = 0.7

Therefore:

0.625 < 0.7

This confirms:

5:8 < 7:10

Cross multiplication avoids decimal conversion when exact integer arithmetic is easier.

Combining Ratios With a Shared Quantity

Suppose:

A:B = 2:3

and:

B:C = 4:5

To combine them, make the B terms equal.

First ratio:

A:B = 2:3

Second:

B:C = 4:5

LCM of:

3 and 4

is:

12

Scale first ratio by 4:

A:B = 8:12

Scale second by 3:

B:C = 12:15

Therefore:

A:B:C = 8:12:15

Another Combined Ratio Example

Suppose:

X:Y = 3:5

and:

Y:Z = 2:7

Make Y equal.

LCM of:

5 and 2

is:

10

Scale:

X:Y = 6:10

and:

Y:Z = 10:35

Therefore:

X:Y:Z = 6:10:35

The shared middle quantity must represent the same amount in both ratios before combining them.

Ratio and Proportional Scaling

If:

A:B = 3:4

and the first quantity becomes:

18

the scale factor is:

18/3 = 6

Therefore the second quantity becomes:

4 × 6 = 24

So:

18:24 = 3:4

This is the basic mechanism behind proportional enlargement and reduction.

Ratio in Recipes

Suppose a mixture uses ingredients in ratio:

2:3:1

and you need:

24 total cups

Total ratio parts:

2 + 3 + 1 = 6

One part:

24/6 = 4 cups

Quantities:

2 × 4 = 8

3 × 4 = 12

1 × 4 = 4

Therefore the mixture uses:

8 cups, 12 cups, and 4 cups

Ratio in Maps

Suppose map scale is:

1:50,000

This means:

1 unit on the map

corresponds to:

50,000 of the same units in reality.

If the map distance is:

3 cm

actual distance:

3 × 50,000

= 150,000 cm

Convert:

150,000 cm = 1.5 km

Therefore:

3 cm on the map represents 1.5 km

Ratio in Similar Figures

Suppose corresponding side lengths have ratio:

2:5

If the smaller figure has a side of:

8 cm

the corresponding larger side is:

8 × 5/2

= 20 cm

Therefore:

8:20 = 2:5

Length ratios scale directly.

Area and volume ratios require powers of the linear scale factor and should not be treated as identical to side-length ratios.

Ratio and Percentages

A ratio can be converted to a percentage depending on the reference.

For:

A:B = 3:4

the first quantity as a percentage of the second is:

3/4 × 100%

= 75%

But if 3:4 represents two parts of one total, then the first category is:

3/(3+4)

= 3/7

of the whole.

Percentage:

3/7 × 100%

≈ 42.86%

These are different questions.

Ratio as a Growth Multiplier

Suppose a value changes from:

80 to 100

The new-to-original ratio is:

100:80

Simplify:

5:4

Numerically:

5/4 = 1.25

Therefore the new value equals:

1.25

times the original, corresponding to:

25%

growth.

Ratios often provide the multiplier underlying percentage-change calculations.

Ratios in Geometric Sequences

A geometric sequence has a constant ratio between consecutive nonzero terms.

For:

5,15,45,135,…

calculate:

15/5 = 3

45/15 = 3

135/45 = 3

Therefore:

common ratio = 3

The next term is:

135 × 3 = 405

Ratios and Arithmetic Sequences

An arithmetic sequence is based on a constant difference rather than a constant ratio.

For:

5,10,15,20,…

differences are:

5,5,5

but ratios are:

2

1.5

4/3

Therefore the sequence is arithmetic, not geometric.

This distinction prevents confusing additive and multiplicative patterns.

Ratios and Remainders

A ratio is based on division, but it should not be confused with a remainder.

For integer division:

a = bq + r

where:

0 ≤ r < b

the quotient q describes how many complete copies of b fit into a, while r is the leftover amount.

For example:

17 ÷ 5 = 3 remainder 2

The ratio:

17/5 = 3.4

is the exact quotient as a rational value.

The remainder:

2

is instead the leftover after three complete groups of five.

These are related through division but represent different mathematical quantities.

Ratio of Remainder to Divisor

In some applications, the remainder itself may be compared with the divisor.

For:

17 ÷ 5

remainder:

2

divisor:

5

Ratio:

2:5

or:

2/5

This equals:

0.4

which is the fractional part beyond the whole-number quotient:

3 + 2/5 = 3.4

This connection shows how ratio, fraction, quotient, and remainder can describe the same division from different perspectives.

Ratios and Real Numbers

Ratios need not always use integers.

For example:

π:2

has numerical quotient:

π/2

which is a real number but not rational.

Likewise:

√2:√8

simplifies:

√2/(2√2)

= 1/2

so a ratio formed from irrational quantities can sometimes simplify to a rational result.

Ratios and Proportions

Two ratios are proportional when they are equal:

a:b = c:d

meaning:

a/b = c/d

For nonzero denominators:

ad = bc

For example:

4:6

and:

10:15

are equivalent because:

4 × 15 = 60

6 × 10 = 60

This is the foundation of proportional reasoning.

Ratio Tables and Constant Multipliers

Suppose:

x:y = 3:8

Then:

Each pair has:

x/y = 3/8

The table can be generated by multiplying both ratio terms by the same scale factor.

Finding an Unknown From a Total and Ratio

Suppose two quantities have ratio:

5:7

and their total is:

144

Total ratio parts:

5 + 7 = 12

One part:

144/12 = 12

First quantity:

5 × 12 = 60

Second:

7 × 12 = 84

Therefore:

The quantities are 60 and 84

Finding a Total From One Ratio Part

Suppose:

A:B = 3:5

and:

A = 27

Since:

3 ratio units = 27

one ratio unit is:

27/3 = 9

Therefore:

B = 5 × 9

= 45

Total:

27 + 45

= 72

So:

B = 45 and total = 72

Finding a Ratio From Actual Quantities

Suppose three quantities are:

24, 36, 60

Their ratio is:

24:36:60

Find the common GCF:

12

Divide every term:

2:3:5

Therefore:

24:36:60 = 2:3:5

Ratios With Zero

A ratio may have zero as the first term:

0:5

which corresponds to:

0/5 = 0

But a ratio:

5:0

cannot be interpreted as an ordinary quotient because:

5/0

is undefined.

Context sometimes uses colon notation informally with zero counts, but division-based calculations require a nonzero second term.

Negative Ratios

Algebraically, ratios may involve negative quantities.

For example:

-2:5

has quotient:

-2/5

And:

-2:-5

has quotient:

2/5

In physical applications, whether negative ratio terms are meaningful depends on what the quantities represent.

Common Mistake: Reversing the Requested Order

If there are:

7 adults

and:

4 children

adults to children:

7:4

children to adults:

4:7

Read the wording in the same sequence in which the ratio should be written.

Common Mistake: Confusing Part-to-Part and Part-to-Whole

With:

3 red

and:

2 blue

red to blue:

3:2

red to total:

3:5

These are different comparisons.

Always determine what the second quantity in the ratio is supposed to represent.

Common Mistake: Using Different Scale Factors

From:

2:3

multiplying only the first term by 4 and the second by 5 gives:

8:15

This is not equivalent.

To preserve a ratio, use the same scale factor:

2×4 : 3×4

= 8:12

Therefore:

2:3 = 8:12

Common Mistake: Simplifying Before Converting Units

For:

2 m : 50 cm

do not simplify numerical values:

2:50

first.

Convert to matching units:

200 cm : 50 cm

Then:

4:1

The numerical terms only become directly comparable after unit conversion.

Common Mistake: Assuming Ratios Must Add to 100

Ratio terms do not normally sum to 100.

For:

3:7

the terms sum to:

10

To convert them into percentages of a whole:

3/10 = 30%

7/10 = 70%

The ratio itself remains:

3:7

Common Mistake: Adding Ratios Directly

If:

A:B = 2:3

and:

B:C = 4:5

you cannot immediately write:

A:B:C = 2:3:5

because the two versions of B do not match.

First equalize the shared quantity.

The correct combined ratio is:

8:12:15

How to Check Equivalent Ratios

Suppose:

14:21

is claimed equivalent to:

10:15

Simplify:

14:21 = 2:3

10:15 = 2:3

Therefore the ratios are equivalent.

Cross multiplication also gives:

14 × 15 = 210

21 × 10 = 210

The claim is correct.

Frequently Asked Questions

What are ratios?

Ratios are multiplicative comparisons between two or more quantities.

How is a ratio written?

Common forms include:

a:b

a to b

a/b

Does order matter in ratios?

Yes. a:b generally differs from b:a.

How do you simplify ratios?

Divide every term by the same greatest common factor.

Simplify 18:30.

GCF = 6

so:

18:30 = 3:5

What are equivalent ratios?

Ratios obtained by multiplying or dividing all corresponding terms by the same nonzero factor.

What is a part-to-part ratio?

A comparison between two categories within one total.

What is a part-to-whole ratio?

A comparison between one category and the total.

Can ratios have more than two terms?

Yes.

For example:

2:3:5

Can ratios contain decimals?

Yes. They can often be scaled into whole-number form.

Can ratios contain fractions?

Yes. Multiplying all terms by a common denominator often simplifies them.

What is the difference between ratios and proportions?

A ratio is a comparison. A proportion states that two ratios are equal.

What is a unit ratio?

A ratio expressed relative to one unit of the second quantity.

How do you divide a total using a ratio?

Add the ratio terms, divide the total by that sum, then multiply one-part value by each ratio term.

Final Example

Three departments receive a budget in the ratio:

4:5:7

The total budget is:

320,000

First calculate total ratio parts:

4 + 5 + 7 = 16

One part:

320,000/16

= 20,000

First department:

4 × 20,000

= 80,000

Second:

5 × 20,000

= 100,000

Third:

7 × 20,000

= 140,000

Therefore the allocation is:

80,000 : 100,000 : 140,000

Check the total:

80,000 + 100,000 + 140,000

= 320,000

Check the ratio by dividing each amount by:

20,000

giving:

4:5:7

The central ratio principles are:

Ratio = one quantity ÷ another

Equivalent ratios use the same scale factor

Multi-part ratios divide a total according to the sum of their ratio units

Once the order, units, and reference quantities are identified correctly, ratios provide a compact way to compare, scale, distribute, and relate numerical quantities.