Fractions – Definition, Properties, Types, Operations & Examples
Factions represent parts of a whole, with a numerator (the top number) indicating how many parts you have and a denominator (the bottom number) representing the total number of equal parts the whole is divided into. Simple fractions have integers in both the numerator and denominator, while complex fractions have fractions in either the numerator or denominator, or both. Proper fractions have numerators smaller than denominators.
Let’s learn about fractions in detail, including their parts, types, properties, and operations.
Table of Content
- What are Fractions?
- Fractions Definition
- Parts of Fraction
- Numerator
- Denominator
- Properties of Fractions
- Types of Fractions
- Unit Fraction
- Proper Fraction
- Improper Fraction
- Mixed Fraction
- Like and Unlike Fractions
- Like Fractions
- Unlike Fractions
- Equivalent Fractions
- Operations on Fractions
- Fraction on Number Line
- Value of Pi in Fraction
- Solved Examples on Fractions
- Fractions Practice Problems
What are Fractions?
If any object is divided in some number of equal parts and some of these parts are taken at once, then fraction is defined as the ratio of the number of taken parts to the total number of divided parts.
For example, a pizza is divided into 8 equal parts, and one part is taken out, thus the fraction representing the taken out part is 1/8 as one part is taken out of 8 equal parts. 1/8 part of the pizza can be represented in many forms such as:
- One-Eighth
- 1/8
- 1 by 8
Fractions Definition
Fractions are numerical expressions used to represent parts of a whole or ratios between quantities. They consist of two numbers separated by a horizontal line, where the number above the line is called the numerator and the number below the line is called the denominator.
The remaining parts collectively will be 7/8 of the pizza as of now; there are only 7 pieces left. Similarly, if we take three parts of this pizza then the fraction representing the taken part will be 3/8 and if 2 pieces are taken out, they will be 2/8 of the entire pizza. It can be simplified and written as 1/4.
Parts of Fraction
There are two parts to any fraction, numerator and denominator. The number at the top is known as the numerator and the bottom number is known as the denominator. If we divide anything into some equal parts then:
Numerator
The numerator represents how many parts of the whole are taken and it is the upper part of the fraction. For example, 3 is the numerator in 3/5, 4 is the numerator in 4/11, 13 is the numerator in 13/7, etc.
Denominator
The denominator represents the total number of equal parts of the whole and is the lower part of the fraction. For example, 5 is the denominator in 3/5, 11 is the denominator in 4/11, 7 is the denominator in 13/7, etc.
Properties of Fractions
There are some important properties of fractions similar to whole numbers, natural numbers, etc. Let’s take a look at those properties:
Property | Description | Example |
---|---|---|
Commutative Property (Addition & Multiplication) | The order of adding or multiplying fractions doesn’t change the result. | Addition: a/b + c/d = c/d + a /b Multiplication: a/b × c/d = c/d × a /b |
Associative Property (Addition & Multiplication) | The way fractions are grouped in addition or multiplication doesn’t change the result. | Addition: (a/b + c/d )+ e/f = a/b + (c/d )+ e/f ) Multiplication: (a/b × c/d) × e/f = a/b × (c/d × e/f) |
Identity Element | For multiplication, the identity is 1 (multiplying a fraction by 1 gives the same fraction). For addition, the identity element is incorrectly noted as 0 in the statement; it should be that adding 0 to a fraction doesn’t change its value. | Multiplication: (a/b) ×1 =a/b Addition: a/b+0 =a/b |
Multiplicative Inverse | The reciprocal of a fraction, when multiplied by the original fraction, gives 1. | (a/b) × (b/a) = 1 |
Distributive Property | Multiplying a fraction by the sum of two fractions equals the sum of each multiplied separately. | a/b × (c/d + e/f) = a/b × c/d + a/b × e/f |
Types of Fractions
There are majorly four types of fractions. They are unit fractions, proper fractions, improper fractions, and mixed fractions. They are categorized on the basis of their numerator and denominator.
Let’s learn about all four types in detail.
Unit Fraction
A unit fraction is defined as a fraction with 1 as a numerator and is known as a Unit Fraction, for example – 1/8, 1/10, 1/4, 1/6, 1/11, etc. It can be said that all unit fractions are proper fractions since all unit fractions have 1 in the numerator, which is lesser than the denominator.
The below-given example explains unit fractions very nicely. Here, a cake is divided into 4 equal parts. If the whole cake is 1, then each quarter of the cake is the fraction 1/4.
Proper Fraction
A Proper fraction is defined as a fraction in which the numerator value is less than the denominator value then. It is known as a proper fraction, for example, 4/9, 1/10, 2/5, 3/7, 5/9, etc.
Improper Fraction
An Improper fraction is defined as a fraction in which the numerator value is greater than the denominator value, then it is known as an improper fraction, for example, 6/5, 11/10, 11/5, 5/3, 2/1, etc.
Mixed Fraction
A Mixed fraction consists of a whole number with a proper fraction then, it is known as a mixed fraction. For Example, if 2 is a whole number and 1/4 is a fraction, then 2¼ is a mixed fraction.
Like and Unlike Fractions
Other types of fractions are on the basis of groups. They involve like and unlike fractions and equivalent fractions. All three types of fractions are explained below in complete detail.
Like Fractions
Any two or more fractions whose denominators are the same are known as like fractions. In other words, those fractions which can be added or subtracted together without taking the lcm of denominators are called like fractions.
Some examples of like fractions are as follows:
- 2/9, 3/9, 5/9, 9/9, 4/9 (Here, the denominators of all the fractions are the same, that is, 9).
- 3/10, 7/10, 1/10, 9/10, 6/10 (Here, the denominators of all the fractions are the same, that is, 10).
- 1/7, 2/7, 4/7, 5/7, 7/7, 8/7.
- 1/2, 7/2, 6/2, 5/2, 9/2.
- 7/5, 1/5, 4/5, 3/5.
Unlike Fractions
The fractions whose denominators are different are called, unlike fractions. In other words, those fractions which can’t be added or subtracted together without taking the lcm of denominators are called, unlike fractions.
Some examples of, unlike fractions are:
- 9/2, 1/6, 5/4, 7/3 (Here, the denominator of all the fractions is different).
- 1/2, 1/4, 2/3, 5/6, 8/9 (Here, the denominator of all the fractions is different).
- 3/8, 2/3, 3/5, 2/7.
- 1/9, 2/7, 3/4, 2/5, 3/2.
- 4/2, 1/6, 2/3, 7/5.
Equivalent Fractions
Equivalent fractions are defined as those fractions which result in the same value after simplification, and then they are equivalent to each other. Solving equivalent fractions can be done by either multiplying both the numerator and denominator by the same number or dividing the numerator and denominator by the same number. Below are some examples of equivalent fractions:
Operations on Fractions
Operations on fractions are simple arithmetic operations like addition, multiplication, division, etc. Let’s take a look at the operations that we can do on fractions:
Operation | Description | Example | Solution |
---|---|---|---|
Reduction of Fractions | Simplifying fractions by dividing the numerator and denominator by the same value. | Reduce 88/42 | 88/42 = 44/21 |
Multiplication of Fractions | Multiply the numerators and the denominators of the fractions. Simplify if possible. | Multiply 21/5 by 35/12 | 21/5 × 35/12 = 49/4 |
Addition of Fractions | Add fractions with a common denominator. If denominators differ, find a common denominator. | Add 4/5 and 7/3 | 4/5 + 7/3 = 47/15 |
Division of Fractions | Flip the second fraction and multiply. Simplify if possible. | Divide 35/20 by 5/10 | 35/20 ÷ 5/10 =5/2 |
Subtraction of Fractions | Subtract fractions with a common denominator. If denominators differ, find a common denominator. | Subtract 1/2 from 3/5 | 1/2−3/5=−1/10 |
Fraction on Number Line
Fractions on a number line are represented between the interval of two integers. Fractions are a part of the whole. Therefore, the whole is divided into equal parts, the number of parts between the integers is decided by the denominator value, and the numerator value is the point where the fraction lies.
For example, if we are required to represent 1/8 on a number line, the division will be between 0 and 1, and since 8 is in the denominator, the whole is divided into 8 equal parts, where the first represents 1/8. Similarly, the second part represents 2/8, and so on.
Value of Pi in Fraction
Pi (π) is a mathematical constant that is defined as the ratio of a circle’s circumference to its diameter. The value of pi in decimal is approximately equal to 3.14. The value of pi (π) in fractions is approximated to 22/7 which is the most popular approximation of π. Other than 22/7, 355/113, 3 1/7, 104348/33215, etc. are also some lesser-known fractional approximations of π. The value of pi is used both in decimal and fraction form based on the type of question given.
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Solved Examples on Fractions
Example 1: Write two equivalent fractions of 3/39.
Solution:
Equivalent fraction by multiplying with the same number, lets multiply by 2:
(3 × 2)/(39 × 2)
= 6/78
Equivalent fraction by division with the same number, here, both numerator and denominator are divisible by 3, dividing by 3:
(3 ÷ 3)/(39 ÷ 3)
= 1/13
Example 2: In a class of 90 students, 1/3rd of the students do not like cricket. How many students like cricket?
Solution:
Fraction of students that do not like cricket = 1/3
Fraction of student that like cricket = 1 – 1/3
= (3 – 1)/3
= 2/3rd students like cricket.
Number of students that like cricket = 2/3 × 90
= (2 × 30)
= 60
Therefore, 60 students like cricket.
Example 3: What type of fraction is this – 1/2, 1/5, 1/7, 1/10, 1/3?
Answer:
This is a Unit fraction because all the fractions have 1 as a numerator.
Example 4: If a recipe needs 3/4 cup of sugar and you want to make half the recipe, how much sugar do you need?
Answer:
Sugar needed for recipe = 3/4 cup
Suger needed for half the recipe = 1/2 of 3/4
Required sugar = 1/2 × 3/4 = 3/8
Therefore, we need 3/8 cup of sugar.
Fractions Practice Problems
1. Add the fractions 2/5 and 3/4.
2. Subtract 3/8 from 7/12.
3. Multiply 1/3 by 7/4.
4. Divide 5/6 by 2/3.
Summary – Fractions
Fractions represent parts of a whole, with a numerator (the top number) indicating how many parts you have and a denominator (the bottom number) representing the total number of equal parts the whole is divided into. Simple fractions have integers in both the numerator and denominator, while complex fractions have fractions in either the numerator or denominator, or both. Proper fractions have numerators smaller than denominators.
FAQs on Fractions
What are Fractions?
A fraction is the ratio of two natural number where first part of the ratio is called numerator and second part of the ratio is called denominator, and these two parts are seperated by a horizontal line.
What is Proper Fraction?
A proper fraction is defined as a fraction in which the numerator valur is smaller than the denominator value. For example, 3/5, 5/7, 11/13, 9/17, etc.
What is Improper Fraction?
An Improper fraction is defined as a fraction in which the numerator value is greater than the denominator value, then it is known as an improper fraction, for example, 7/5, 13/10, 9/5, 17/3, 5/1, etc.
What is Mixed Fraction?
A mixed fraction is a combination of a whole number and proper fraction. For example, 2 1/2 (2 whole 1. by 2), 7 2/5 (7 whole 2/5), etc.
How to Compare Fractions?
We compare fractions is done to understand which fraction is larger or which fraction is smaller among the given fractions. If the denominator of the fractions given are same, only the numerators are compared, if the fractions have different denominators, first the LCM of fractions are taken and denomiantoras are made equal, and then the numerators are compared.
For example, 3/19 and 7/19 when compared, it can be observed that the denominator are same, therefore, numerators are compared, 3 > 7, therefore, 7/19 > 3/19.
How to Multiply Fractions?
Multiplication of fraction do not require any common denominator. Here, simply the numerator is multiplied by numerator of the other fraction and denominator is multiplied by the denominator. It can also be simplified afterwards if the fraction can be reduced. Fractions can also be simplified while multiplying with each other. For example: 3/7 × 1/2 = 3/14.
How to Divide Fractions?
Division of fractions do not require any common denominators, the simplest way to divide fractions is to flip the fraction after division sign, that is, reciprocate the fraction, and change the sign from division to multiplication. Now, simply apply the multiplication of fractions rules and solve. For example: 4/9 ÷ 2/3 = 4/9 × 3/2 = 2/3.
How to Add Fractions?
Adding fractions is possible only when the denominator are same, incase the denominators of the fractions are not same, it is important to bring them to a common denominator. Find the LCM of fractions in order to find the common denominator. For example: 3/4 + 1/2 = 3/4 + 2/4 = 5/4.