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### Types and Operations of Fractions

The fraction is a very important concept in maths. It is defined as part of a whole thing. For example half of a cake or a quarter of a pizza. It is represented in the form of 1/2. In 1/2, 1 is called the numerator while 2 is called the denominator. The numerator is used to describe the part of the entity which is in discussion while the denominator is used to represent the total number of parts.

## Types of the fraction

**1 What is Proper Fraction?**

A proper fraction has a numerator that is smaller than its denominator, for example, 3/7. A proper fraction is always less than 1.e.g 4/5,6/7 etc.

**2 What is an Improper Fraction?**

An improper fraction has a numerator that is bigger than its denominator, for example, 10/7. An improper fraction is always greater than 1. e.g 16/7,13/9 etc. *If the numerator is equal to denominator then the fraction is equal to 1. These *are also called improper fractions e.g. 5/5=1,4/4=1 etc.

### 3 What is a Mixed Fraction?

A fraction that is represented as 2 1/4 is called a mixed fraction because it has a whole number and a fraction together.

## Comparing Fractions

**Common fractions.**

A fraction in which numerator and denominator both are integers is called a common fraction. e.g. 5/7 is a common fraction while 1/2/3/5 is not common fraction here numerator and denominator are also a fraction i.e 1/2 and 3/5.

**Equivalent Fractions.**

Some fractions may look different but are really the same, so these are called equivalent fractions. for example:

^{4}/_{8} |
= | ^{2}/_{4} |
= | ^{1}/_{2} |

(Four-Eighths) | (Two-Quarters) | (One-Half) |

It is always desirable to show an answer in its simplest possible way. In these above examples, it is 1/2. This process is called ** simplifying or reducing a fraction**.

#### 1 Least Common Denominators(LCD).

In Maths, the lowest common denominator or **least common denominator** (abbreviated **LCD**) is the least common multiple of the denominators of a set of fractions. LCD of a set of a fraction is the lowest number which we can use in the denominators to convert these fractions into equivalent fractions that have the same denominators.

e.g.

1/3 and 4/5 are two fractions. In order to create the same denominator of both fractions, we convert them into their equivalent fractions in such a way that both of them have a common denominator. We find the LCD of fractions to find equivalent fractions. In the above example, 5/15 and 12/15 are equivalent to 1/3 and 4/5. if we find their Least Common Multiplication (LCM) which became their LCD and it is used to create the Equivalent Fractions.

#### 2. Least Common Multiplication (LCM).

A common multiple is a number that is a multiple of two or more numbers. The common multiples of 3 and 4 are 0, 12, 24,… The **least common multiple** (**LCM**) of two numbers is the smallest number (not zero) that is a multiple of both. So the LCM of 3 and 4 is 12.

## Basic Math Operations.

** How to do the addition of fractions?**

1-**With Common Denominator**: We add the numerators and then simply it at the end to get the addition e.g

A)2/7+4/7+5?7=11/7.

B) 5/8+1/8=6/8=3/4. (divide by 2 to simplify it.)

The addition of fractions with common denominators can be practiced here using our worksheets for year 4.

All our worksheets are designed on the basis of the latest national curriculum.

2-**With different Denominators**: If denominators of the fractions are different then we use the following method.

e.g. 3/8+1/4=??

Method: We must *somehow* make the denominators the same.

In this case, it is easy, because we know that ^{1}/_{4} is the same as ^{2}/_{8} :

^{3}/_{8} |
+ | ^{2}/_{8} |
= | ^{5}/_{8} |

If we cannot find the suitable equivalent fraction we have to use the following method to find the common denominators and then add them later.

#### How to Find the Addition of fractions by the LCD method?

- First, convert all the integer and mixed numbers into proper fractions.
- Then for all the denominators, find the LCM(Least Common Multiplication).
- This LCM number will be an LCD set of fractions.
- Now divide the LCD found in step 3 by each of the denominators.
- After that multiply both the numerator and denominator by the number found in step 4.
- In the end, add the numerators of the newly found fractions from step three but keep the same denominators.
- This will give the result of the addition of a set of fractions.

*Example.*

Question 1.Add the following.

4/5+ 6/7=?

Answer.

- First, find the LCM of both denominators which are 35.
- Now Use this LCM to make LCD and divide 35 by 5 and get 7. Then divide 35 by 7 and get 5.
- After that multiply both numerator and denominator separately by their respective number. In this case, multiply 7 by 4 and then by 5 e.g. 4*7/5*7=28/35. Similarly 6*5/7*5=30/35.
- Add the new numerators of both fractions by keeping the denominators the same to get the result. i.e 28/35+30/35=
**58/35**.answer.

### 2 How to do subtraction of fractions?.

The method of subtracting the fractions is the same as addition. If denominators are the same then subtract the numerators like simple subtraction e.g. 5/4-3/4=2/4=1/2.

If denominators are not equal then first make them equivalent fraction using the LCD method of equivalent fractions.

### 3- How to multiply fractions?

To multiply fractions:

- First Simplify the fractions if it is not in the lowest terms.
- Then Multiply the numerators of the fractions to get the new numerator.
- Then Multiply the denominators of the fractions to get the new denominator. Simplify the resulting fraction if possible.

For Example 5/6× 3/8.

Step 1. First Multiply both numerators(top numbers). 5× 3 =15

Step 2. Then Multiply both denominators(bottom numbers). 6× 8=48.

Step 3. Then write them in the form fraction and simplify if required. For example 15/48.

### 4 Dividing Fractions.

*Turn the second fraction upside down, then multiply. *There are 3 Simple Steps to Divide Fractions:

Step 1. Turn the second fraction *(the one you want to divide by)* upside down

(this is now a reciprocal).

Step 2. Multiply the first fraction by that reciprocal.

Step 3. Simplify the fraction (if needed).