Fractions are among the many basic mathematical concepts that are used frequently in real life. This is why we teach the concept of fractions to school-going students very early! Without knowing fractions, it is difficult to make your way through life, and especially to indulge in activities of any sort. Fractions are also probably the first mathematical concept taught to any student, which requires using a particular syntax. At the age at which students are taught this concept, understanding such a syntax can be an incredibly challenging task!
This is where the issue with the teaching of fractions props up often. While the initial imparting of education about fractions can be a challenging task, what is even more problematic is teaching students operations involving fractions. There are multiple methods to add, subtract, multiply, and divide different types of fractions, each of which is of use at different times. This article aims to answer a few questions about fractions, especially in the context of their subtraction. Some of these questions are listed below.
What are the different types of fractions?
What are the different operations that can be conducted on fractions?
What are the ways in which fractions can be subtracted?
What are the applications of the subtraction of fractions?
Due to the different syntax in which fractions are generally written, there are many different types of fractions. The following are the three main types of fractions:
These are the most common types of fractions that you will encounter in your daily life. Proper fractions are the type of fractions in which the numerator is less than the denominator. The most common place where you will find these types of fractions is when percentages are expressed in the form of fractions. In any context where a part of the whole is being said, proper fractions are used.
Examples: ¾, ⅓, ⅖
These types of fractions are generally more scarce in daily life. The fractions in which the numerator of the fraction is greater than the denominator are known as improper fractions. These fractions are known as improper since it is very unusual to express in fraction form a quantity that is greater than the number of parts that it has. For example, expressing 9/7 in words would mean saying '9 out of 7 parts', which sounds absurd and somewhat inconvenient. This is why the usage of improper fractions will always be less than the other two types of fractions. These types of fractions are also known as top-heavy fractions.
Examples: 3/2, 4/3, 9/5
In straightforward terms, this is just a much better way of writing an improper fraction. When you express a quantity in the form of an improper fraction, what you are saying is that the amount is greater than the whole. In such a circumstance, you can take the whole out of the fraction. You can then express the fraction in the form of a whole number and a proper fraction. These types of fractions are used freely and regularly. For example, when you say 'one and a half', you are using an improper fraction. Even when you use terms of time such as 'six and a half hours', you are using a mixed fraction.
Examples: 1½, 3⅓, 5⅔
Fractions, at their core, are also numbers. This means that all the operations that can be performed on numbers such as integers, whole numbers, and natural numbers can also be performed on mixed fractions, improper fractions, and proper fractions. What is different is the method of performing these operations. The table method that is used for the addition, subtraction, and multiplication of whole numbers, and the long division method that is used for their division, is not possible for fractions. The processes for conducting these operations can be as easy or more complex. This depends on the type of fractions that these operations are being worked on.
The easiest method of subtraction of fractions comes into the picture when the fractions that are being subtracted have the same denominator. Having the same denominator brings uniformity to the question, which makes the operation considerably easy. In real life, subtracting with the same denominator is like subtracting from the same pie. The number of pieces is the same; you simply take away the number of parts you want.
To subtract fractions with the same denominator, you subtract one numerator from the other. The denominator should be kept constant. The result will have the difference between the numerators as the resultant numerator. The common denominator will be the resultant denominator.
Formula: Answer = (Numerator 1 – Numerator 2)/(Denominator)
Example: 3/7 – 1/7
= (3 – 1)/7
The cross multiplication method is the most commonly used method when it comes to subtracting fractions with different denominators. To use a real-life example, when you are subtracting fractions with unlike denominators, you are no longer taking pieces of the same pie. You are attempting to subtract a portion of one pie from a portion of another. Both have been divided into a different number of pieces. This is bound to be more than a little arduous.
In the cross multiplication methods, the first step is to multiply the numerator of each fraction with the denominator of the other. Now, add the resultant two numbers. This is the numerator of your new fraction. Now, multiply the denominators of the two fractions. This is the denominator of your new fraction.
Formula: ((Numerator 1 x Denominator 2) – (Numerator 2 x Denominator 1))/(Denominator 1 x Denominator 2)
Example: 5/9 – 3/7
= ((5 x 7) – (3 x 9))/(9 x 7)
= (35 – 27)/63
This is an easy method of subtracting fractions, but this method is applicable only when one of the denominators is a multiple of the other. Though you might not find a lot of use of this method, this is a quick trick method that has several real-life applications.
To use this method, the initial step is to check whether the two denominators are divisible. If one denominator is a multiple of the other, divide the large denominator by the smaller one. The resultant number is your multiplying factor. Now, take the fraction with the smaller denominator. Multiply its numerator as well as its denominator by this factor. You will realise that you now have two fractions with the same denominator. Now, subtract the two numerators to get the numerator of your results. Let the denominator be the same.
Formula: ((Numerator 1) – (Numerator 2 x (Denominator 1/Denominator 2)))/(Denominator 1)
Example: 5/80 – 3/20
= ((5) – (3 x (80/20)))/(80)
= (5 – 12)/80
Using this method requires knowledge of the concept of the lowest common multiple. This is the oldest and more traditional method for the subtraction of one fraction from another. To carry out this operation, the first step is to conduct a prime factorisation of the denominators of both the fractions that are to be subtracted. Now, find out the common factors of both the denominators. Now, take the non-common factors of Numerator 1 and multiply them to Numerator 1 and Denominator 1, and take the non-common factors of Numerator 2 and multiply them to Numerator 2 and Denominator 2. The resultant two fractions you will get will have the same denominator. To get the result, subtract the numerators from each other and let the denominator remain the same.
Example: 4/75 – 2/30
75 = 3 x 5 x 5
30 = 2 x 3 x 5
LCM = 3 x 5 x 2 x 5 = 150
5/75 = 8/150; 2/30 = 10/150
= 8/150 – 10/150
This article gives you several methods in your next step to learn numbers using Cuemath. It provides an introduction to the concept of fractions. The reader is introduced to different types of fractions and where these fractions are generally used. There are many operations that can be carried out on fractions, especially when it comes to the subtraction operation. There are different ways in which the subtraction operation may be carried out on fractions, both with the same denominator and different denominators. And voila, the reader may now be called a pro at fraction subtraction!