Know the importance of Fractions. Refer to types of fractions and their usage for various numbers. Follow the definition of fraction numbers along with properties, real-life examples. Check important points to convert fractions into decimals along with the different types of fractions. For better understanding, we took few examples and explained all of them in detail.

## Fractions – Definition and Types

Fractions are the most important concept in mathematics. They are defined as the whole parts. Fraction value is nothing but a section or portion of a quantity. These are denoted by the symbol “/”.

**Example:** \(\frac { 4 }{ 6 } \) is a fraction value.

In the above fraction, 4 is defined as the numerator and 6 is defined as the denominator.

### Quick Links of Fractions Concepts

Below is the list of topics that comes under the topic Fractions. If you have any queries on the concerned topic you can simply tap on the links available and learn the fundamentals involved in it easily. Get a good hold of the concept by practicing on a consistent basis.

- Types of Fractions
- Equivalent Fractions
- Like and Unlike Fractions
- Conversion of Fractions
- Fraction in Lowest Terms
- Addition and Subtraction of Fractions
- Multiplication of Fractions
- Division of Fractions

### History of Fraction

Fraction word is derived or originated from the Latin word “Fractus” which means broken. To consider a real-life example, suppose the cake is a whole part, if we cut some portion say \(\frac { 1 }{ 4 } \), it is defined as the fraction portion of the cake.

### Definition of Fractions

A numerical value is represented by a fraction which means the portion or parts of a whole. For suppose, if a number(a) is to be divided into four parts, then it is represented by \(\frac { a }{ 4 } \). Therefore, the number \(\frac { a }{ 4 } \) defines \(\frac { 1 }{ 4 } \)th of the number a. Fraction values are an important part of daily life.

### Types of Fractions

There are three types of Fractions that are used majorly. The three types of fractions are proper, improper, and mixed fractions which are majorly used. Apart from these, there are also other fractions – like fractions, unlike fractions, equivalent fractions. Therefore, there are totally 6 types of fractions.

- Proper Fractions
- Improper Fractions
- Mixed Fractions
- Like Fractions
- Unlike Fractions
- Equivalent Fractions

The above-mentioned first 3 fractions define single fractions whereas the other three fractions define the comparison of two or more fractions.

These fractions play an important role in real life as all the measured quantities cannot be absolute whole numbers, dealing with some portions or parts is also necessary, that is where the concept of fractions has come in.

#### 1. Proper Fractions

The proper fractions are those in which the numerator value is less than the denominator value. i.e., Numerator < Denominator. After further simplification, the value of the proper fraction will always be less than 1.

**Example:** \(\frac { 5 }{ 9 } \) will be the proper fraction

#### 2. Improper Fractions

The improper fractions are those in which the numerator value is greater than the denominator value. i.e., Denominator < Numerator. After further simplification, the value of the improper fraction will always be greater or equal to 1 but not less than 1.

**Example:** \(\frac { 9 }{ 5 } \) will be the improper fraction

#### 3. Mixed Fractions

The mixed fraction is nothing but the combination of an integer value and a proper fraction. These are also called as mixed numerals or mixed numbers. These mixed fractions can be converted to a normal fraction. Mixed fractions are always greater than 1.

**Example:** 3 \(\frac { 3 }{ 2 } \) is a mixed fraction.

#### 4. Like Fractions

As the name defines like fractions are those which are the same or alike. i.e., the fractions with the same denominators are called the like fractions.

**Example:**

\(\frac { 7}{ 2 } +\frac { 5 }{ 2 }+\frac{3}{2}+ \frac {1}{2} =\frac { 16 }{ 2 } \) = 8

#### 5. Unlike Fractions

As the name defines, unlike fractions are those which are different or unequal. i.e., the fractions with unlike or unequal denominators are called, unlike fractions.

**Example:** \(\frac { 1 }{ 5 } \),\(\frac { 1 }{ 4 } \),\(\frac { 1 }{ 3 } \),\(\frac { 1 }{ 2 } \)

**Steps to Simplify Unlike Fractions:**

Simplifying the unlike fractions is a little bit of a lengthy process.

- First, factorize the denominator and simplify them.
- For example, if we have \(\frac { 1 }{ 2 } \) and \(\frac { 1 }{ 3 } \). Then, find the solution of LCM of 2 and 3 which equals 6.
- Now, multiply the first equation i.e., \(\frac { 1}{ 2 } \) with 3, and second equation i.e., \(\frac { 1 }{ 3 } \) with 2, the multiplication implies for both numerator and denominator.
- After applying the multiplication, the result will be \(\frac { 3 }{ 6 } \) and \(\frac { 2 }{ 6 } \).
- Now add both the values of \(\frac { 3 }{ 6} \) and \(\frac { 2 }{ 6 } \).
- The result value will be \(\frac { 5 }{ 6 } \).

#### 6. Equivalent Fractions:

Equivalent Fractions are those whose result value is the same after the simplification process and they represent the same portion or quantity of the whole. Therefore, the fractions which are equal to each other are known as equivalent fractions.

**Example:** \(\frac { 2 }{ 3 } \) and \(\frac { 4 }{ 6 } \) are equivalent fractions

Since \(\frac { 4 }{ 6 } \) can be written as \(\frac { (2*2) }{(2*3) } \)

The result value will be \(\frac { 2 }{ 3 } \) which means that both are equivalent.

### Properties of Fractions

In similar to whole numbers and real numbers, fractions also holds some of the important properties.

- Associative and Commutative Properties holds true for multiplication and addition of fractions.
- Fractional multiplication is always 1 and the identity element of the fractional addition is 0.
- If a and b are non-zero elements, then the multiplicative inverse of \(\frac { a }{ b } \) is \(\frac { b }{ a } \) where a and b must be non-zero elements.
- The distributive property of multiplication over addition is applicable to all fractional numbers.

### Important Points about Fraction Types

Proper fraction value after the simplification of the equation is always less than 1.

Every natural number can be represented in fraction form, where the denominator value is equal to 1

Improper Fraction value after the simplification of expression is always greater than or equal to 1, but no less than 1.

Every mixed fraction can be converted into a normal fraction.

Every improper fraction can also be converted into a mixed fraction.

A mixed fraction simplification always results in greater than 1.

If the numerator value is equal to 1, then that fraction is called a unit fraction.

**Example:**

One-fifth of the whole – \(\frac { 1 }{ 5 } \)

One-fourth of the whole – \(\frac { 1 }{ 4 } \)

One-third of the whole – \(\frac { 1 }{ 3 } \)

One half of the whole – \(\frac { 1 }{ 2 } \)

### Fractional Rules

There are some rules to be followed to solve fraction based problems.

**Rule 1:** Before going for the addition or subtraction of fractions, make sure that the denominators are the same or equal. Therefore, only the fractions with the same denominator can be added or subtracted.

**Rule 2:** To multiply the fractions, the numerators and the denominators are multiplied separately, and later the simplification of fractions is done.

**Rule 3:** When dividing a fraction from another, we have to find the reciprocal of that fraction and then multiply it with the first fraction to find the answer.

#### Addition of Fractions

If two fractions have the same denominator, then adding those fractions will be easy. We just add the numerators and give the common denominator.

**Example:**

The above is the case where both the denominators have the same value.

If the denominators have different values, then we have to simplify the equation by finding the LCM of denominators and then making that common for both the fractions.

**Example:**

\(\frac { 3 }{ 4 } \)+\(\frac { 2 }{ 3 } \)

The denominators of the equation are 4 and 3

LCM of the denominators 4,3 is 12

Then multiply the first equation by \(\frac { 4 }{ 4 } \) and the second equation by \(\frac { 3 }{ 3 } \)

Hence, we get

= \(\frac { 9 }{ 12 } \)+\(\frac { 8 }{ 12 } \)

= \(\frac { (9+8) }{ 12 } \)

= \(\frac { 17 }{12 } \)

#### Subtraction of Fractions

If two fractions have the same denominator, then subtract the numerator numbers and get the final result.

**Example:**

If two fractions have different denominators, then take the LCM of the numerators and then making it common for 2 fractions.

**Example:**

\(\frac { 2 }{ 3 } \)–\(\frac { 3 }{ 4 } \)

In the above equation, the denominators are 3,4

The LCM of 3,4 is 12

Therefore, multiply the first equation by \(\frac { 4 }{ 4 } \) and the second equation by \(\frac { 3 }{ 3 } \), we get

\(\frac { 8 }{ 12 } \) –\(\frac { 9 }{ 12 } \)

Now that the denominators are the same, subtract the first numerator value from the second.

= \(\frac {(8-9)}{ 12 } \)

=\(\frac { -1 }{ 12 } \)

#### Multiplication of Fractions

To multiply two fractions, both the numerators from both equations and both the denominators from both equations are multiplied.

**Example:**

Multiply \(\frac { 3 }{ 7 } \) and \(\frac { 2 }{ 3 } \)

= \(\frac { (3×2) }{ (7×3) } \)

=\(\frac { 6 }{ 21 } \)

=\(\frac { 2 }{ 7 } \)

#### Division of Fractions

To divide two fractions, we need to multiply the first fraction with the reciprocal of the second fraction.

**Example:**

Divide \(\frac { 3 }{ 7 } \) and \(\frac { 2 }{ 3 } \)

To divide two equations, multiply the first fraction with the reciprocal of the second fraction.

=\(\frac { (3×3) }{(7×2) } \)

= \(\frac { 6 }{ 14 } \)

= \(\frac { 3 }{ 7 } \)