A fraction consists of two numbers (a/b), a numerator and a denominator. The number written at the top is the numerator, and the one written at the bottom is the denominator. Mathematicians came up with three major types of fractions, named as proper fractions (or the fractions less than one and greater than 0, with numerator less than denominator), improper fractions (or the fractions more than one or equal to one with a numerator greater than or equal to the denominator), and mixed fractions (a combination of a whole number and a proper fraction). This article contains a brief description of the improper fractions and performing various operations on them.
The term ‘fraction’ represents a quantity that is part of an entire object. We will understand fractions with an example. Suppose we have an outsized cake and that the cake is divided into 8 equal slices. Then each portion of the slice is merely 1/8th of the total quantity of the cake. Here, 1/8 = fraction.
Numerator: The top half of a fraction that represents the number of parts you have got. Within the above example, 1 is the numerator.
Denominator: The bottom half of a fraction represents the amount into which the entire object is split. Within the above example, 8 is the denominator.
Line: The lines that divide the numerator and denominator.
The number of the shape a/b, where a and b are natural numbers, is referred to as a fraction. In a fraction a/b, a is understood as the numerator and b is understood as the denominator.
Fraction is a part of the whole.
The numerator and the denominator of any of the numbers are called its terms.
Example: 7/10 may be a fraction, where the numerator is 7, and therefore, the denominator is 10.
Types of Fractions:
A fraction is a part of any aggregate out of the entire aggregate, where the entire aggregate can be any number, a selected value, or a thing. Each and every aggregate that comes in our day-to-day measurements can’t be an absolute integer. So we have to deal with parts of a whole or portions of an entire or fraction.
A fraction can be divided into two parts. The upper part is called numerator, while the lower part is called denominator. Based on the numerator and denominator, the types of fractions are proper and improper fractions, mixed fractions, equivalent fractions, and like and unlike fractions.
When the numerator of a fraction is smaller than its denominator, it is called a proper fraction. In other terms, the proper fraction is in which this condition satisfies:
Numerator < Denominator
When the denominator of a fraction is smaller than its numerator, it is called an improper fraction. In other terms, the improper fraction is in which this condition satisfies:
Numerator > Denominator
Remember one more thing here, that you can represent any of the natural numbers in the form of a fraction with denominator 1. A proper fraction will always be less than 1, while an improper fraction will be greater than 1 or equal to 1.
Mixed fraction definition: A fraction comprising a number and a fraction is named as a mixed fraction.
You can easily convert a mixed fraction into an improper fraction, and vice versa. A mixed fraction will always be greater than 1.
Fractions which have the same denominators are called like fractions. For Example, the fractions 2/7, 3/7, 5/7, and 6/7 have the same denominator, 7. Hence, these are like fractions.
Simplification of like fractions is very easy. For example, if you would like to feature the above four fractions, all you have to try to do is add the numerators. The denominator will remain an equivalent.
Example of operation in like fraction:
2/7 + 6/7 + 3/7 + 5/7 =
(2+6+3+5)/ 7 =
Fractions which have different denominators are called unlike fractions. For example, ⅔ and ¼ have different denominators. So they are categorised as unlike fractions. Simplification of unlike fractions is not straight-forward as like fractions.
Addition of Unlike Fractions
For example, to perform operations on the above two, unlike fractions, first, we have to convert them into like fractions. The steps involved are as follows:
First, calculate the LCM of two denominators, 3 and 4.
For example, LCM of 3 and 4 will be 12. This LCM will be the common denominator of both fractions.
Now, calculate the value of the first fraction, (⅔). To solve this, divide the LCM of these two numbers, which ( from step 2) is calculated in the previous step. Now, divide the LCM of these two numbers by the denominator of the first fraction, 12/ 3 = 4. Now, multiply 4 by the numerator, which gives 8. Therefore, the first fraction, after converting into a like fraction, is 8/12.
Similarly, calculate the like fraction for the second fraction, which is ¼ . To do it, divide the LCM calculated in the first step, which is 12, by the denominator of the second fraction.
So, 12 ÷ 4 = 3. Now, multiply 3 by the numerator (1), which gives 3. Therefore, the second fraction becomes 3/12. Now, both the fractions have the same numerator, i.e., 12.
Now add the two like fractions in the similar manner as we have done in the previous section.
So, 8/12 + 3/12 = ( 8+3) / 12 = 11/12 . So, ⅔ + ¼ = 11/12.
Fractions, which after simplification give the same value, are called equivalent fractions. For example, ⅓ and 30/90 are equivalent fractions as both of them give the same number of simplifications.
It is a fraction whose numerator is one, and therefore, the denominator is a positive integer, which is named a unit fraction. Samples of unit fractions are ½, ⅕, ⅛, etc.
Identifying a Fraction Greater Than 1 or Less Than 1
Whenever the numerator of the fraction is lesser than the denominator, the fraction is lesser than one.
On the other hand, whenever the numerator of the fraction is greater than the denominator, it is a fraction more than 1.
And then, there are mixed fractions too. Mixed fractions contain both a whole number and a proper fraction, like 8 ¼.
Students who were successful in this have already generalised the rule: fractions greater than 1 have numerators larger than their denominators, those that are less than 1 have numerators smaller than their denominators.
Let’s determine if each fraction is greater than 1 or less than 1.
A Brief on a Fraction Greater Than 1
There are several number line fractions greater than 1. Here is a simple example to help you understand more about its representation.
Let us consider the below example of three whole pizzas (each divided into four parts) and a pizza from which one piece is taken away.
Now, you can represent the number of pieces of pizza with you in the form of fractions. The denominator here will be the total number of pieces making a pizza, which is 4. Now the total pieces of pizza in all are 15, representing the numerator.
This gives us the fraction 15/4, which is an improper fraction.
Conversion of a Fraction Greater Than 1 into a Mixed Fraction
For converting a fraction more than one into a mixed fraction, here are the steps that you must follow:
Divide the numerator of the given fraction by its denominator.
First, write the whole number answer received (which is also the quotient of the division).
Now write the fraction with the remainder as the numerator and the denominator staying the same.
Convert 13/4 to a mixed fraction.
Solution: Divide the numerator by the denominator, 13/4 = 3, with a remainder of 1. This can also be written as 3 R 1.
Now write two and then write the remainder 1 above the denominator 4.
= 3 *1/4
Conversion of a Mixed Fraction into a Fraction More Than 1
Here are the steps that you must follow to convert a mixed fraction into an improper fraction:
Multiply the whole number part of the given fraction by the denominator of the fraction.
Add the resultant to the numerator.
Then, write the resultant of the sum on the numerator part and keep the denominator the same as earlier.
Convert 3 ⅘ into an improper fraction.
Solution: Multiply the whole number part of the fraction by the denominator
3 x 5 = 15.
Now add the resultant to the numerator.
15 + 4 =19.
Write the resultant part above the earlier denominator: 19/5
Representation of Fractions on a Number Line Greater Than 1
To represent a fraction greater than one on a number line, follow the mentioned steps:
Write a fraction greater than 1 in the form of a mixed fraction.
Now start from the whole number part of the mixed fraction.
Divide the section between the mentioned whole number and the following whole number into equal parts, as stated in the denominator.
Now count the parts for the number as mentioned in the numerator.
Mark that on the number line to complete the representation.
Q1: Convert 17/3 into a mixed fraction.
Ans. 17/3 = 17 3 =5 with a remainder 2.
= 5 R 2
= 5 ⅔
Q2: Convert 2 4/9 into an improper fraction.
Ans. Multiply the whole number part of the fraction by the denominator.
2 x 9 = 18.
Now add the resultant to the numerator.
18 + 4 = 22.
Write the resultant part above the earlier denominator: 22/9