1. Fractions
The term ‘Fraction’ represents the parts of a whole object or collection of objects. Let us understand this concept using an example. A cake that is divided into 12 equal parts. Now, if you want to express one selected part of the cake, we can express it as 1⁄12 showing that out of 12 equal parts, we are referring to 1 part.
It can be read as
- One-twelfth, or
- 1 over 12
1.1 Parts of a Fraction
A fraction has two components. The number on the top of the line is called the numerator. It tells how many equal parts of the whole object or collection of objects are taken. The number below the line is called the denominator. It shows the total number of equal parts the whole object is divided into or the total number of the same objects in a collection. All fractions consist of a numerator and a denominator and they are separated by a horizontal bar known as the fractional bar.
For example, in the fraction 7⁄9, 7 is the numerator and 9 is the denominator.
1.2 Types of Fractions
Based on the values of the numerator and denominator, there are different types of fractions as mentioned below:
Proper Fractions
The fraction in which the numerator is less than its denominator is called Proper Fraction. For example, 6/7, 7/8, 4/5, etc are proper fractions.
Improper Fractions
The fraction in which the numerator is more than or equal to its denominator is called an Improper Fraction. It is always the same or greater than the whole. For example, 4/3, 7/2, 9/5, and so on.
Unit Fractions
The fraction in which the numerator is 1 is known as Unit Fraction. For example, 1/9, 1/17, 1/3, and so on.
Mixed Fractions
A mixture of a whole number and a proper fraction is known as Mixed Fraction. For example, 3 4⁄5, here 3 is the whole number and 4⁄5 is the proper fraction.
Like Fractions
The fractions having the same denominator are known as Like Fractions. For example, 5/7 and 4/7 are like fractions. Here, we have divided the whole into 7 equal parts.
Unlike Fractions
The fractions having different denominators are known as Unlike Fractions.
For example, 4⁄7 and 8⁄11 are unlike fractions.
Equivalent Fractions
The fractions that represent the same value after they are simplified are known as Equivalent Fractions. We can use the following method to find equivalent fractions of any given fraction:
- We can multiply or divide both the numerator and the denominator of the given fraction by the same number as required.
Example: Find two fractions that are equivalent to 4⁄7.
Solution:
To find the first equivalent fraction, we multiply both the numerator and the denominator of
4⁄7 by 2 i.e., 4⁄7 = 4⁄7 × 2⁄2 = 8⁄14
To find the second equivalent fraction, we multiply both the numerator and the
denominator of 4⁄7 by 3 i.e., 4⁄7 = 4⁄7 × 3⁄3 = 12⁄21
Hence, 8⁄14, 12⁄21 and 4⁄7 are all equivalent fractions.
1.3 Fractions on a Number Line
A number line is a straight line with numbers placed at equal intervals along its length. The fractions on a number line can be represented by making equal parts of a whole. For example, if we need to represent 1⁄10 on the number line, we need to mark 0 and 1 on the two ends and divide the number line into 10 equal parts. Then, the first interval can be marked as 1⁄10. Similarly, the next interval can be marked as 2⁄10, the next one can be marked as 3⁄10, and so on. It should be noted that the last interval represents 10⁄10 which means 1. Observe the following number line that represents these fractions on a number line.
1.4 Conversion between Mixed Fraction and Improper Fraction
Example: Convert 3 5⁄7 into an improper fraction.
Solution:
Example: Convert 25⁄8 into a mixed fraction.
Solution:
8 goes into 25 three times with one as the remainder.
25⁄8 = 3 1⁄8
1.5 Arithmetic Operations involving Fractions
Addition and Subtraction of Fractions:
- If the fractions have the same denominator, add or subtract the numerators.
- If the fractions have different denominators, first find the lowest common multiple of the denominators, convert the fractions with LCM as the common denominator, and then add or subtract the numerators.
Example: Find 3⁄5 + 1⁄5
Solution:
Since the denominators are the same, simply add the numerators: 3⁄5 + 1⁄5 = 3⁄5+1⁄5 = 4⁄5
Example: Find 2⁄3 - 1⁄7
Solution:
Find the LCM of the denominators : LCM of 3 and 7 is 21
Make 21 the common denominator : 2⁄3 × 7⁄7 = 14⁄21 and 1⁄7 × 3⁄3 = 3⁄21
Subtract the fractions : 14⁄21 - 3⁄21 = 14⁄21 - 3⁄21 = 11⁄21
Multiplication of Fractions:
- To multiply fractions, multiply the numerators together and multiply the denominators together.
Example: Find 3⁄5 × 6⁄7
Solution:
3⁄5 × 6⁄7 = 18⁄35
Division of Fractions:
- To divide two fractions, keep the first fraction as it is.
- Replace '÷' by '×'
- Turn the second fraction upside down and multiply.
Example: Workout 1⁄5 ÷ 4⁄5
Solution:
1⁄5 ÷ 4⁄5 = 1⁄5 × 5⁄4 = 1⁄4
1.6 Fractions of Amount
- To work out a fraction of an amount, divide the amount by the denominator and multiply it by the numerator or vice versa.
Example: Calculate 5⁄8 of 200.
Solution:
Divide 200 by 8 and then multiply by 5.
5⁄8 of 200 = (200 ÷ 8) × 5
= 25 × 5
= 125
