Proper Fractions

Before we learn about proper fractions, we must understand what fractions are. Fractions can be defined as parts of a whole. For instance, if we consider cutting a cake into 4 pieces, then each piece can be expressed as a fraction as one-fourth, i.e., ¼ parts of the entire cake. Similarly, 2 pieces would make 2/4 or ½ portion. Here, the number of parts or portions are called numerators and the total equal number of parts the whole is divided into is called the denominator. Now proper fractions are the fractions where the numerator is smaller than the denominator.   

Fraction Notation

A fraction has two parts:

Numerator – The top number of a fraction is the numerator of the fraction. It shows the number of parts we are considering out of a whole.

Denominator - The bottom number of a fraction is the denominator of the fraction. It shows the number of equal parts that a whole is divided into.

For example - ½

In the above example, 1 is considered as the numerator and 2 is considered as the denominator of the fraction.

What is a Proper Fraction?

A proper fraction is a fraction that has a smaller numerator compared to its denominator. It is called a proper fraction owing to its proper adherence to the definition of a fraction and the need for a fraction. To understand this better, let us discuss how they are different from improper fractions. 

Improper fractions are the ones where the denominator is smaller than the numerator. If we recall the definition of fractions, we will realise that if a denominator is smaller than the numerator in a fraction, then the total value will be greater than 1, which makes things complicated. It must be noted here that if a fraction is greater than 1, then the fraction can be divided into a whole number part and a fractional part. For instance, 5/2 means two and a half, which can be written as 2½, which is called a mixed number. This implies that 5/2 comprises 2 whole parts and half of one whole part. Hence, 5/2 is said to be improper as it can be written appropriately as a whole number and a proper fraction (½).   

Examples of Proper Fractions

• 4 out of 6 equal pizza slices are an example of a proper fraction since the fraction can be expressed as 4/6 or 2/3, where the numerator is smaller than the denominator. 

• 80 out of 100 marks in an exam can be an example of a proper fraction as the number of marks scored is less than the total number of marks. The fraction can be expressed as 80/100, and its reduced form can be expressed as ⅘. 

• 3 out of 20 students in a group can be an instance of proper fraction, where the number of students in consideration (3) is smaller than the total number of students in the group (20). The fraction can be expressed as 3/20.  

• 110 pages out of 530 in a book is an example of a proper fraction, which can be expressed as 110/530 or 11/53. Here, the number of pages in consideration is less than the total number of pages in the book, that is, the numerator is less than the denominator. 

Solved Examples

Here are some proper fraction examples of addition and subtraction with different denominators:

1. 4/5 + 2/3

Solution: Here, the denominators are different.

So, we will find the LCM to make the denominators equal.

LCM of 5 and 3 is 15.

Accordingly,

4 x 3/ 5 x 3 = 12/15

2 x 5 / 3 x 5= 10/15

Now, we will add both numbers with similar denominators, i.e.,

= 12/15 + 10/15

= 22/15

2. 1/4 -1/5

Solution: Here, the denominators are different.

So, we will find the LCM to make the denominators equal.

LCM of 4 and 5 is 20.

Accordingly,

1 x 5 /4 x 5 = 5/20

1 x 4 / 5 x 4 =4/20

= 5/20 -4/20

=1/ 20

Quiz Time

1.  Which of the following is considered as a proper fraction?

1. 2/5

2. 8/7

3. 1/1

4. 10/9

2. Which of the following is an addition of 7/12 and 3/12?

1. 4/6

2. 21/12

3. 4/12

4. 5/6

3. What fraction of the numbers from 2 to 12 are prime numbers?

1. 1/11

2. 10/11

3. 5/11

4. 6/11

Fun Facts

• The word fraction is acquired from the Latin word ‘Fractus’, which implies  ‘broken’.

• The fraction originated from the Egyptian era, which is renowned as one of the oldest civilizations in the world. However, fractions are considered as numbers. They are actually used to compare whole numbers with one another.

Conclusion

Proper fractions can be considered the appropriate way for writing a fraction that is simply a fraction, without a whole number included. Proper fractions are used with whole numbers to denote a sum that is greater than 1.