Proper Fraction: Definition, Examples and Types

What Is a Proper Fraction?

A proper fraction is a fraction in which the numerator is smaller than the denominator.

The numerator is the top number of a fraction, while the denominator is the bottom number.

For example:

• 2/3

• 4/5

• 5/8

• 7/10

• 9/11

All of these are proper fractions because the numerator is smaller than the denominator.

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Rule for a Proper Fraction

Numerator < Denominator

A proper fraction always has a value less than 1.

For example:

3/5 < 1

because 3 is smaller than 5.

Proper Fraction Definition

The definition of a proper fraction is:

A proper fraction is a fraction whose numerator is less than its denominator. Its value is always less than 1.

For example, 7/9 is a proper fraction because:

• Numerator = 7

• Denominator = 9

• 7 < 9

Therefore, 7/9 is a proper fraction.

Proper Fraction Examples

Here are some examples of proper fractions:

5 Examples of Proper Fractions

Five simple examples are:

1. 1/3

2. 2/5

3. 3/7

4. 4/9

5. 5/6

In each fraction, the numerator is smaller than the denominator.

Read more: Unit Fraction- Definition, Examples, Practice Problems

How to Identify a Proper Fraction

You can identify a proper fraction in three simple steps:

Step 1: Find the numerator

Look at the top number.

Example: In 4/9, the numerator is 4.

Step 2: Find the denominator

Look at the bottom number.

In 4/9, the denominator is 9.

Step 3: Compare the numbers

If:

Numerator < Denominator

the fraction is a proper fraction.

Example:

4 < 9

So, 4/9 is a proper fraction.

Proper and Improper Fractions

Proper and improper fractions are different based on the relationship between the numerator and denominator.

Examples

3/8 is a proper fraction because:

3 < 8

8/3 is an improper fraction because:

8 > 3

5/5 is an improper fraction because:

5 = 5

Difference Between Proper and Improper Fractions

The main difference is the size of the numerator compared with the denominator.

Proper Fraction

In a proper fraction, the numerator is smaller than the denominator.

Example:

5/9

Since 5 < 9, it is a proper fraction.

Its value is less than 1.

Improper Fraction

In an improper fraction, the numerator is equal to or greater than the denominator.

Examples:

9/5
7/7
12/4

Their values are equal to or greater than 1.

Quick Comparison

Proper: numerator < denominator

Improper: numerator ≥ denominator

Is 1/2 a Proper Fraction?

Yes. 1/2 is a proper fraction because the numerator, 1, is smaller than the denominator, 2.

Its value is:

1/2 = 0.5

which is less than 1.

Is 3/4 a Proper Fraction?

Yes. 3/4 is a proper fraction because:

3 < 4

Therefore, its value is less than 1.

Is 5/5 a Proper Fraction?

No. 5/5 is not a proper fraction.

The numerator and denominator are equal:

5 = 5

Therefore, 5/5 is an improper fraction. Its value is exactly 1.

Can a Proper Fraction Be Equal to 1?

No.

A proper fraction always has a numerator smaller than its denominator, so its value is always less than 1.

A fraction such as:

4/4 = 1

is not a proper fraction because the numerator and denominator are equal.

Read more: Vulgar Fraction: Meaning, Steps & Solved Questions

Can a Proper Fraction Be Greater Than 1?

No.

A proper fraction always has a value less than 1.

For example:

3/7 < 1

If a fraction has a value greater than 1, its numerator is greater than its denominator, making it an improper fraction.

Example:

8/5 > 1

Can a Proper Fraction Be Negative?

In basic school mathematics, proper fractions are usually introduced as positive fractions with a numerator smaller than the denominator.

A negative fraction such as -2/5 has a magnitude less than 1, but whether it is called a “proper fraction” depends on the definition being used in a particular textbook or course.

For school-level questions, follow the definition taught in your curriculum.

Proper Fractions on a Number Line

Proper fractions lie between 0 and 1 on a number line.

For example:

1/4, 1/2 and 3/4

all lie between 0 and 1.

The larger the numerator, while keeping the denominator the same, the greater the proper fraction.

For example:

1/5 < 2/5 < 3/5 < 4/5

All four are proper fractions.

Proper Fractions and Whole Parts

A proper fraction represents a part of one whole.

For example, imagine a pizza divided into 8 equal pieces.

If you have 3 pieces, you can represent them as:

3/8

Because 3 pieces are less than all 8 pieces, 3/8 is a proper fraction.

This makes proper fractions useful for describing parts of objects, quantities and groups.

How to Add Proper Fractions

Proper fractions can be added using the same rules as other fractions.

When the Denominators Are the Same

Add the numerators and keep the denominator unchanged.

Example:

2/7 + 3/7 = 5/7

The answer, 5/7, is also a proper fraction because 5 < 7.

When the Denominators Are Different

First find a common denominator.

Example:

1/3 + 1/4

The LCM of 3 and 4 is 12.

Convert the fractions:

1/3 = 4/12

1/4 = 3/12

Now add:

4/12 + 3/12 = 7/12

So:

1/3 + 1/4 = 7/12

How to Subtract Proper Fractions

Same Denominator

Subtract the numerators and keep the denominator the same.

Example:

5/7 - 2/7 = 3/7

Different Denominators

Find a common denominator first.

Example:

3/4 - 1/6

The LCM of 4 and 6 is 12.

Convert:

3/4 = 9/12

1/6 = 2/12

Subtract:

9/12 - 2/12 = 7/12

Therefore:

3/4 - 1/6 = 7/12

How to Multiply Proper Fractions

To multiply fractions, multiply the numerators and denominators.

Example:

2/3 × 4/5

Multiply the numerators:

2 × 4 = 8

Multiply the denominators:

3 × 5 = 15

Therefore:

2/3 × 4/5 = 8/15

The answer is also a proper fraction because 8 < 15.

How to Divide Proper Fractions

To divide one fraction by another, multiply the first fraction by the reciprocal of the second fraction.

Example:

3/4 ÷ 2/5

The reciprocal of 2/5 is 5/2.

So:

3/4 × 5/2 = 15/8

The result is 15/8, which is an improper fraction.

This is an important point: the result of an operation involving proper fractions does not always have to be a proper fraction.

Read more: Numerator and Denominator: Definition and Difference

Proper Fraction and Mixed Fraction

A mixed number contains a whole number and a proper fraction.

For example:

2 1/3

This means 2 whole units and 1/3 of another unit.

A mixed number can be converted into an improper fraction.

Example: Convert 2 1/3 to an Improper Fraction

Step 1: Multiply the whole number by the denominator.

2 × 3 = 6

Step 2: Add the numerator.

6 + 1 = 7

Step 3: Put the result over the original denominator.

7/3

Therefore:

2 1/3 = 7/3

The result is an improper fraction, not a proper fraction.

Proper Fraction Examples with Solutions

Example 1: Is 9/10 a Proper Fraction?

Yes.

Numerator = 9

Denominator = 10

Since:

9 < 10

9/10 is a proper fraction.

Example 2: Is 7/12 a Proper Fraction?

Yes.

7 < 12

Therefore, 7/12 is a proper fraction.

Example 3: Is 11/8 a Proper Fraction?

No.

11 > 8

Therefore, 11/8 is an improper fraction.

Example 4: Is 6/6 a Proper Fraction?

No.

6 = 6

Therefore, 6/6 is an improper fraction and its value is 1.

Example 5: Which of These Are Proper Fractions?

2/5, 7/4, 3/8, 9/9, 5/6

Check each one:

• 2/5 → Proper

• 7/4 → Improper

• 3/8 → Proper

• 9/9 → Improper

• 5/6 → Proper

Therefore, the proper fractions are:

2/5, 3/8 and 5/6

Practice Questions on Proper Fractions

Question 1

Which is a proper fraction?

a) 7/4
b) 3/8
c) 9/9
d) 11/5

Answer: b) 3/8

Question 2

Is 5/6 a proper fraction?

Answer: Yes, because 5 < 6.

Question 3

Is 8/5 a proper fraction?

Answer: No. It is an improper fraction because 8 > 5.

Question 4

Which fraction is proper?

a) 4/4
b) 8/3
c) 2/7
d) 9/5

Answer: c) 2/7

Question 5

What is the value of a proper fraction compared with 1?

Answer: A proper fraction is always less than 1.

Quick Revision

Remember these points about proper fractions:

• A proper fraction has a numerator smaller than its denominator.

• Its value is less than 1.

• 2/5, 3/7 and 5/9 are examples of proper fractions.

• A fraction with an equal or larger numerator is an improper fraction.

• 5/5 is not proper because its value is 1.

• Proper fractions can be added, subtracted, multiplied and divided using the usual fraction rules.

• An operation involving proper fractions can produce an improper fraction.

• A mixed number is not directly converted into a proper fraction. It is normally converted into an improper fraction.