# Performing Arithmetic Operations on Different Types of Fractions

In mathematics, fractions are used to show how many parts the whole number is divided. Fractions are widely used in the number system, algebra, rational numbers, and many other branches of mathematics.

In this article, we will learn about the definition, types, and rules of fractions along with a lot of examples.

## Fraction

A fraction represents a part of a whole or, more generally, any number of equal parts. It is a numerical quantity that is not a whole number e.g., ½, 0.5.

In arithmetic, a fraction is a number expressed as a quotient, in which a numerator is divided by a denominator.

A fraction is a number that represents a part of the whole. When an object or group of objects is divided into equal parts, then each individual part is a fraction. A fraction is basically written as 3/5, 9/11, 23/5, and so on.

## Types of Fractions

There are six main types of fractions. Let us discuss them briefly.

1. Proper Fraction
2. Improper Fraction
3. Mixed Fraction
4. Like Fraction
5. Unlike Fraction
6. Equivalent Fraction

### 1. Proper Fraction

When the numerator is smaller than the denominator, it is said to be a proper fraction. These fractions are less than one and none of them lies beyond one on a number line. The denominator represents the number of equal parts in which the whole is divided. Proper fractions are written as 2/5, 5/7, 9/13, and so on.

### 2. Improper Fraction

When the numerator is greater than the denominator, it is said to be an improper fraction. These fractions are greater than one and lie beyond one on the number line. The denominator represents the number of equal parts required. The numerator is the number of objects available. Improper fractions are written as 7/5, 9/7,19/13, and so on.

### 3. Mixed Fraction

A number that is written as the product of a whole number with a proper fraction is known as a mixed fraction. In simple words, it is the combination of natural numbers and fractions. It is basically an improper fraction. Written as 1-2/5, 2-5/7, 9-10/23, and so on.

An improper fraction can be expressed as a mixed fraction by dividing the numerator by the denominator and obtaining the quotient and remainder.

### 4. Like Fraction

The fractions which have the same denominator are called the like fractions. e.g., 1/4, 3/4, 5/4, 7/4 are like fractions.

The calculation of such fractions is easy, as all the denominators here are the same. Suppose we need to subtract all the above like fractions, then; 1/4 – 3/4 – 5/4 – 7/4 = (1-3-5-7)/4 = -14/4 = -7/2

### 5. Unlike Fraction

The fractions which do not have the same denominators or different denominators are called, unlike fractions. e.g., 2/3, 3/4, 1/5 are unlike fractions.

Calculations of such fractions are rather lengthy as first we have to make all the denominators the same by making multiple and selecting the least common multiple and then simplifying.

### 6. Equivalent Fraction

When two or more fractions have the same results after simplification for which they represent the same portion of the whole, then such fractions are equal to each other and are called equivalent fractions. e.g., 1/2, and 2/4, are equivalent, 1/3, and 3/9 are equivalent.

## Arithmetic Operations on Fractions

Addition and subtraction of fractions is an easier method when you are dealing with integers. Let us take some examples to understand this method of fractions. You can also check the calculation of fractions by an online Fraction Calculator.

Example 1 – Add the fractions 1/2, 3/4, 1/6?
SOLUTION –
Step 1: place addition signs between the fractions.
1/2 + 3/4 + 1/6
These are unlike fractions first change them in like fractions
Step 2: Take the denominators of the fraction.
2, 4, 6
Step 3: Find the LCM of these numbers.
Multiples of 2 = 2, 4, 6, 8, 10, 12, 14, …
Multiples of 4 = 4, 8, 12, 16, 20, 24, ….
Multiples of 6 = 6, 12, 18, 24, 30, ….
Common multiples of 2, 4 and 6 = 12, 24, …
LCM of 2, 4 and 6 = 12
Step 4: Make the denominator of each fraction equal to 12.
(1/2)(6/6) = 6/12
(3/4)(3/3) = 9/12
(1/6)(2/2) = 2/12
Step 5: Now add the like fractions
6/12 + 9/12 + 2/12 = (6+9+2)/12
= 17/12
Example 2 – Subtract the fractions 2/3, 4/6, 5/9?
SOLUTION –
Step 1: Place subtraction signs between the fractions.
2/3 – 4/6 – 5/9
Step 2: take the denominators of the fraction.
3, 6, 9
Step 3: Find the LCM of these numbers.
Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, …
Multiples of 6 = 6, 12, 18, 24, 30, ….
Multiples of 9 = 9, 18, 27, 36, 45, ….
Common multiples of 3, 6 and 9 = 18, 36 …
LCM of 3, 6 and 9 = 18
Step 4: Make the denominator of each fraction equal to 18.
(2/3)(6/6) = 12/18
(4/6)(3/3) = 12/18
(5/9)(2/2) = 10/18
Step 5: Now subtract the like fractions
12/18 – 12/18 – 10/18 = (12-12-10)/18
= -10/18
= -5/9

### Multiplication of fractions

Multiplication is also simple infractions as we have to multiply the numerators with the numerator and the denominator with the denominator. Let us take some examples to understand this method of fractions.

Example – Multiply the fractions 1/2, 3/4, 2/6?
SOLUTION –
Step 1: place multiplication signs between the fractions.
1/2 x 3/4 x 2/6
Step 2: multiply all the numerators.
1x3x2 = 6
Step 3: Multiply all the denominators.
2x4x6 = 48
Step 4: Write the results in numerator and denominator.
1/2 x 3/4 x 2/6 = 6/48
Step 5: Simplify
1/2 x 3/4 x 2/6 = 6/48 = 1/8

### Division of Fractions

Division of fractions is also an easy way to solve the fractions as in this method we have to change the division sign in a multiplication sign by reciprocal the next term as making the numerator a denominator and denominator a numerator. Let us take some examples to understand this method of fractions.

Example – Divide the fractions 1/2, 2/6?
SOLUTION –
Step 1: place the division sign between the fractions.
(1/2) / (2/6)
Step 1: Take the reciprocal of the second fraction to replace the division sign with multiplication.
= 1/2 x 6/2
Step 2: multiply all the numerators.
1×6 = 6
Step 3: Multiply all the denominators.
2×2 = 4
Step 4: Write the results in numerator and denominator.
1/2 x 6/2 = 6/4
Step 5: Simplify
1/2 x 6/2 = 6/4 = 3/2

## Summary

Now you are witnessed that this topic is very simple. Once you grab the basic concepts of this topic you can easily solve the problems related to fractions. Fractions are widely used in mathematics. Fractions are simply called rational numbers.