A fraction is a number that describes a relationship between a part and a whole.

Where the numerator represents part and the denominator represents whole.

A fraction can be used to name a part of a whole.

Concept of Half

Half: The fraction "half" represents one of two equal parts of a whole. It is denoted by the fraction 1/2. The numerator, 1, signifies that we have one part out of two equal parts, while the denominator, 2, indicates the total number of equal parts that make up the whole.

Types of Fractions

Proper Fractions

A fraction is considered proper when its numerator is smaller than its denominator.

Example: 3/5, 1/4

Improper Fractions

A fraction is considered improper when its numerator is greater than its denominator.

Example: 5/3, 8/3

Mixed Fractions

A combination of a whole number and a proper fraction is known as a mixed fraction.

Example: 2^{1}⁄_{5}, 6^{3}⁄_{2}

Unit Fractions

When the numerator is 1, the fraction is called a unit fraction.

Example: 1/2 and 1/7

Like Fractions

When the denominator of the given fractions are same, we call the fractions as like fractions.

Example: 3/5 and 2/5

Unlike Fractions

When the denominator of the given fractions are different, we call the fractions as unlike fractions.

Example: 3/7 and 2/5

Operations on Fractions

There are four basic operations that can be performed on numbers. They are addition, subtraction, multiplication and division.

Addition of Fractions

The addition is one of the fundamental operations that are also applicable to fractions. There are several methods of addition of fractions. Let us know about them.

Addition of Like Fractions: We know that the like fractions have identical denominators. To add the like fractions, add the values of the numerators, keeping the denominators the same.

Example: 2/5 + 1/5

2/5 + 1/5 = (2 + 1) / 5 = 3/5

Addition of Unlike Fractions: We know that unlike fractions have different denominators. So, we need to convert the unlike fractions into like fractions. It means the fractions must have identical denominators.

To add the unlike fractions, we need to follow some steps:

1. Find the LCM of the denominators of the given unlike fractions. 2. Change the denominators into the obtained LCM. This process can change the numerators of the given, unlike fractions. 3. Now, add the numerators.

Example:

Subtraction of Fractions

We have discussed the addition of fractions. Similarly, we can subtract fractions. Let us see the subtraction of like and unlike fractions.

Subtraction of Like Fractions: Subtraction of like fractions are similar to the addition of the like fractions. To subtract the like fractions, subtract the values of the numerators, keeping the denominators the same.

Example: 4/5 – 1/5

4/5 – 1/5 = (4 -1)/5 = 3/5

Subtraction of Unlike fractions: Here, we will follow the same steps which we followed in the addition of unlike fractions. But instead of adding numerators, we will subtract them.

Let us learn the steps:

1. Find the LCM of the denominators of the given unlike fractions. 2. Change the denominators into the obtained LCM. This process can change the numerators of the given unlike fractions. 3. Now, subtract the numerators.

Example:

Multiplication of Fractions

The numerators and denominators of two fractions are multiplied separately when they are multiplied. The first fraction’s numerator will be multiplied by the second’s numerator, and the first fraction’s denominator will be multiplied by the second’s denominator. In the end, we will reduce the fraction to its lowest form if it is required.

Example:

Division of Fractions

Dividing the fraction by another fraction is multiplying the first fraction by the reciprocal of the second fraction.

Let us learn the following steps of the division of fractions.

1. We will keep the first fraction the same, and we need to determine the reciprocal of the second fraction. 2. Change the division sign by a multiplication sign and multiply the first fraction with the reciprocal of the second fraction. 3. Find the simplest form of the fraction, if needed.

Example:

Identification and usage of signs >, < and =

Now let's explore the identification and usage of the signs (>), (<), and (=) when comparing fractions.

Greater Than (>): The symbol ">" is used to compare fractions when the numerator of one fraction is larger than the numerator of another, while the denominators remain the same.

Example: Consider the fractions 3/8 and 2/8. Here, 3 is greater than 2. Thus, we can write: 3/8 > 2/8

When comparing fractions, if the numerator of one fraction multiplied by the denominator of the other fraction is greater than the numerator of the second fraction multiplied by the denominator of the first fraction, then the first fraction is greater than the second fraction.

Example: Compare 3/4 and 2/5

Since 15 is greater than 8, we can conclude that 3/4 is greater than 2/5.

Less Than (<): The symbol "<" is used to compare fractions when the numerator of one fraction is smaller than the numerator of another, while the denominators remain the same.

Example: Let's compare the fractions 1/5 and 3/5. In this case, 1 is smaller than 3. Therefore, we can write: 1/5 < 3/5

When comparing fractions, if the numerator of one fraction multiplied by the denominator of the other fraction is less than the numerator of the second fraction multiplied by the denominator of the first fraction, then the first fraction is less than the second fraction.

Example: Compare 1/3 and 5/8

Since 8 is less than 15, we can conclude that 1/3 is less than 5/8.

Equal (=): The symbol "=" is used to compare fractions when the numerators and denominators of two fractions are equal, signifying that they represent the same value.

Example: If we compare the fractions 2/9 and 2/9, we can see that both the numerators and denominators are the same. Thus, we can write: 2/9 = 2/9

When comparing fractions, if the numerator of one fraction multiplied by the denominator of the other fraction is equal to the numerator of the second fraction multiplied by the denominator of the first fraction, then the fractions are equal.

Example: Compare 4/9 and 8/18

Since 72 is equal to 72, we can conclude that 4/9 is equal to 8/18.

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