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Fractions

Introduction to Fraction

Fractions are numbers that represent a part of the whole. When an object or a group of objects is divided into equal parts, then each individual part is a fraction. A fraction is usually written as 1/2 or 5/12 or 7/18 and so on. It is divided into a numerator and denominator where the denominator represents the total number of equal parts into which the whole is divided. The numerator is the number of equal parts that are taken out. For e.g. in the fraction 3/4, 3 is the numerator and 4 is the denominator.

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Real Life Example of a Fraction

It’s your birthday and mom has ordered pizza for you and your friends. When the pizza arrives, you open the box and find that it is cut into slices. Let’s assume that there are 8 slices and you have 7 friends. So, there are 8 people who are going to eat the 8 slices of the pizza.

fraction

How much does each person get? Well, if we divide the entire pizza into eight equal parts, then each person gets 1/8 or one-eighth of the pizza. The pizza can be cut into a different number of equal slices creating different fractions. (Like a 6-slice pizza or a 4-slice pizza or a 12-slice pizza)

Fractions on a Number Line

Fractions can also be shown on a number line just like whole numbers. Let’s try and plot 1/2 on a number line. Now, we know that 1/2 is greater than 0 but lesser than 1. Hence, it lies between 0 and 1. Further, since the denominator is 2, we divide the distance between 0 and 1 into two equal parts. Refer diagram below:

fraction

Let’s look at one more example. How can we show 2/3 on a number line? Again, we know that 2/3 is greater than 0 but less than 1 (since the numerator is smaller than the denominator). Next, since the denominator is 3, we divide the distance between 0 and 1 into three equal parts. Now, 2/3 is two parts out of these three parts as shown below:

fraction

Proper, Improper and Mixed Fractions

In a fraction there are two simple possibilities:

  1. The numerator is smaller than the denominator
  2. The numerator is bigger than the denominator

Proper Fraction

When the numerator is smaller than the denominator, it is a Proper Fraction. These fractions are less than 1 and none of them lies beyond 1 on the number line. The denominator represents the number of equal parts in which the whole is divided.

And, the numerator represents the number of these equal parts that are considered. The example of the pizza stated above with each person getting 1/8th of the pizza shows a proper fraction.

Improper Fraction

When the numerator is larger than the denominator, it is an Improper Fraction. These fractions are greater than 1 and lie beyond 1 on the number line. The come into play when more than one object is divided equally into certain parts. The denominator represents the number of equal parts required. The numerator is the number of objects available.

Example

Ram, Rachna, Rohini, and Ravi decide to eat some apples. They go to Ram’s house and his mother hands them a basket of apples. They sit in the garden to eat them. On opening the basket, they find that there are 5 apples inside. How can they share these 5 apples among the four of them?

Ram picks up a knife and cuts all apples into four equal parts. Then, he distributes one part of each apple to everyone. Hence, each of them receives one part of all 5 apples. So, the number of apples per person = 5/4

Mixed Fractions

Taking the same example as above, Rachna suggests that they can take one apple each which leaves them with one extra apple. Now, she divides the apple equally into four parts and gives one part to everyone. Hence, the 5 apples are equally divided among them. To put it in an equation,

Number of apples per person = 1 ¼

This is a Mixed Fraction since it has a combination of a whole and a ‘part’. An improper fraction can be expressed as a mixed fraction by dividing the numerator by the denominator and obtaining the quotient and remainder.

Example

5/4 is an improper fraction. When we divide 5 by 4, we get

  • Quotient = 1
  • Remainder = 1

Hence, the fraction can be written as 5/4 = 1 ¼ or Quotient (Remainder/Divisor)

Some Other Types of Fractions

Equivalent Fractions

Are 1/2, 2/4 and 3/6 equal? Not sure? Let’s find out.

fraction

The above diagram represents 1/2, 2/4 and 3/6. If we were to place all three squares on top of one another, do they look equal? Yes, of course. Hence, they are equal. Such fractions are Equivalent Fractions since they represent the same part of a whole. You can check the equivalence between fractions by calculating the simplest form of each fraction.

A fraction is said to be in the simplest (or lowest) form if its numerator and denominator have no common factor except 1. Find the highest common factor (HCF) between the numerator and denominator and divided it by the HCF. So, 2/4 = (2 x 1) / (2 x 2) = 1/2.

Like Fractions

Like Fractions are fractions with the same denominator. So, 1/12, 3/12, 7/12, and 9/12 are all like fractions.  Since, like fractions have a common denominator, comparing them is relatively simple. Which of these two fractions is bigger? 2/6 or 5/6? In this case, the whole is divided into 6 parts. Now, 5 out of those 6 parts is certainly bigger than 2 parts. Hence, 5/6 > 2/6.

Unlike Fractions

Unlike Fractions are fractions with different denominators. So, 1/5, 5/8, and 9/12 are all unlike fractions. Which of these fractions is bigger? 1/4 or 4/13? In this topic, we will look at the best way to compare fractions. Since unlike fractions have different denominators, we can compare them as follows:

Unlike fractions with the same numerator

Let’s compare 1/3 and 1/5. These are unlike fractions (different denominators) but have the same numerator (1).

  • In 1/3, we divide the whole into 3 equal parts and take 1.
  • In 1/5, we divide the whole into 5 equal parts and take 1.
  • Notice that in 1/3, the whole is divided into a smaller number of parts than in 1/5.
  • Therefore, the equal part that we get in 1/3 is larger than the equal part we get in 1/5.
  • Since in both cases we take the same number of parts (i.e. one), the portion of the whole showing 1/3 is larger than the portion showing 1/5.

Hence, 1/3 > 1/5. So, if the numerator is the same in two unlike fractions, the fraction with the smaller denominator is greater than that with a larger denominator.

Unlike fractions with different numerators

Which of these two fractions is greater? 2/3 or 3/4? In such cases, we try to find equivalent fractions of each of them so that the denominator becomes the same. The equivalent fractions for

2/3 are: 4/6, 6/9, 8/12, 10/15, etc.
3/4 are: 6/8, 9/12, 12/16, 15/20, etc.

From the list above, we can see that the equivalent fractions with the same denominator are: 8/12 and 9/12. Using the like fractions rule, we know that 9/12 > 8/12. Hence, 3/4 > 2/3.

Tip: Find the LCM of the denominators for faster calculations.

Adding and Subtracting Fractions

Like fractions

Adding two or more like fractions are done by following these steps:

  1. Add the numerators of all the like fractions
  2. Retain the denominator (which is common across all like fractions)

So, the sum of 3/7 and 2/7 is, 3/7 + 2/7 = (3+2)/7 = 5/7

Also, subtracting two or more like fractions is done by following these steps:

  1. Subtract the smaller numerator from the bigger
  2. Retain the denominator

So, the difference between 7/9 and 4/9 is, 7/9 – 4/9 = (7-4)/9 = 3/9 or 1/3

Unlike fractions

As we learned in comparing, adding or subtracting unlike fractions is done by finding equivalent fractions with a common denominator and then applying the rules for like fractions. So, the sum of 1/5 and 1/2 is calculated as follows:

Equivalent fractions of:
1/5 are 2/10, 3/15, 4/20, etc.
1/2 are 2/4, 3/6, 4/8, 5/10, etc.

From the list above, we can see that the equivalent fractions with the same denominator are: 2/10 and 5/10. Hence, 1/5 + 1/2 = 2/10 + 5/10 = 7/10. Also, the difference between 1/2 and 1/5 is calculated as follows, 1/2 – 1/5 = 5/10 – 2/10 = 3/10.

Solved Example for You

Question 1: Asha and Samuel have bookshelves of the same size partly filled with books. Asha’s shelf is 5/6th full and Samuel’s shelf is 2/5th full. Whose bookshelf is more full? By what fraction?

Answer 1: 5/6 and 2/5 are unlike fractions. to compare them, we will find their equivalent fractions with a common denominator. The LCM of 6 and 5 (denominators of both the fractions) is 30. Hence, the equivalent fractions with a common denominator are:

(5 x 5) / (6 x 5) and (2 x 6) / (5 x 6)
Or, 25/30 and 12/30

So, Asha’s shelf is 5/6th or 25/30th full and Samuels’s shelf is 2/5th or 12/30th full. So, it is easy to deduce that Asha’s shelf is more full. The difference is as follows: 25/30 – 12/30 = 13/30.

Question 2: What is a Fraction?

Answer: Fractions are numbers representing a part of the whole. When we divide an object or group of them into equal parts, then each individual part is referred to as a fraction. We usually write down fractions as ½ or 6/12 and more. Moreover, it divides into a numerator and denominator.

Question 3: What is the Proper Fraction?

Answer: A proper fraction is when the numerator is smaller than the denominator. They are less than 1 and not a single one of them lie beyond 1 on the number line. Moreover, the denominator is known to represent the number of equal parts in which we divide the whole.

Question 4: What is an Improper Fraction?

Answer: Improper Fraction is on where the numerator is larger than the denominator. Similarly, improper fractions are greater than 1 and on the number line, they lie beyond 1. You can use them when more than one object divides equally into specific parts.

Question 5: How do we add fractions?

Answer: When you need to add two or more fractions, you need to first add the numerators of all the similar fractions then retain the denominator. A denominator is common throughout all the like fractions.

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I need full concept of fractions it means additions, subtraction, division,multiplication, mixed fraction,types of fractions, simplifying fractions.

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