Difference Between Fraction and Rational Numbers

Fraction and rational numbers are the two most commonly used terms in mathematics. Appearing a bit similar to each other, they often confuse people. Although the concepts of these vital mathematical components are related in some aspects, there is a remarkable difference between them. Here, we are available to provide you with a clear idea of rational numbers and fractions along with some examples so that you will be left with no further doubts.  

 

Fraction-

Definition: A fraction or fractional number is a number in the form p/q where p and q are the whole numbers, and q is not equal to zero (0). It expresses a part of the whole or any number of equal parts. It can also be defined as the ratio of two integers, where the upper number (numerator) tells how many parts we have, and the lower one (denominator) shows the number of equal parts into which the whole is divided. In another way to say, a fraction represents a divisional expression in which the divisor and dividend are integers, and the divisor is not equal to zero.  For example, 3/5, 9/6, 8/4, etc., are the fractions of fractional numbers. 


Examples of Different Types of Fractions

In mathematics, fraction or fractional numbers are classified into many types. Here, we are putting light on almost every type of fraction by showing their examples.

  • Proper Fractions: The numerator is always smaller than the denominator. For example, 3/8 and 7/9.

  • Improper fractions: The numerator is always larger than the denominator. Example, 9/2 and 7/5.

  • Mixed Fractions: Composed of a whole number and a fraction. For example, 3 (3/2) and 5 (2/7).

  • Equivalent Fractions: These are the fractions whose numerators and denominators can be divided by the same number. Example, 2/12 = 3/18 and 5/10 = 10/20.

  • Like Fractions: These are fractional numbers with the same denominators. Example, 2/5; 3/5. 

  • Unlike Fractions: These are the fractions with different denominators. Example, 2/3; 15/13.

 

Rational Numbers-

Definition: Rational numbers are those numbers which are in the form of a/b where a and b are integers, and b is not equal to zero (0). It can also be expressed as a ratio of integers, i.e., can be written as a fraction of two integers with the upper number as numerator and bottom as the non-zero denominator. Since the denominator can be equal to 1, all integers are rational numbers. Moreover, several floating-point numbers can be expressed as fractions. Hence, they are also rational numbers. For instance, we can write 1.5 as 3/2, 6/4, 9/6, and more. Accordingly, it is a rational number. 


Examples of Rational Numbers

Rational numbers, in general, can appear in the four forms - integers, whole numbers, natural numbers, and fractions. Based on this information, let's see the examples of rational numbers.

  • Since number 8 can be written as fraction 8/1, it is a rational number.

  • 3/4 is a rational number because we can write it as a fraction 

  • We can write 1.5 as the ratio 3/2. Hence, it is also a rational number

  • O.333...can be written as 1/3. Therefore, it is a rational number

  • Recurring decimals like 0.262626..., all finite decimals, and all integers are also rational numbers.

 

Difference Between Fraction and  Rational Numbers

Hopefully, after going through the above-given statistics regarding fractions and rational numbers, you now can differentiate between these two numbers. Nevertheless, a table that shows some clear differences in fractional and rational numbers is as follows:

 

Fractional Numbers

Rational Numbers

p and q are natural numbers

p and q are integers

All fractions are rational numbers 

Not all rational numbers are fractions. Only the rational numbers where both p and q have positive integers are fractions

Example: 1/2, 25/5, 9/7

Example: 8/1, 5/-8, -23/-27, 100/-23

 

 Examples-

Example 1: At Green Valley School, there are 14 male teachers and 11 female teachers. What fractions of the total number of teachers are female?

Solution: According to the question, 

The numerator (p) of the fraction = the number of female teachers.

The denominator (q) of the fraction = the total teachers in the school. 

So, Fraction of female teachers = number of female teachers/ total number of teachers

                                                     = 11/ (14 + 11) 

                                                     = 11/ 25.

 

Example 2: 2½ is a mixed fraction. Identify whether it is a rational number or not?

Solution: The Simple form of 2½ is 5/2

Where, 

The numerator 5 is an integer

Denominator 2 is also an integer and not equal to zero (0).

So, we can say that yes, 2½ = 3/2 is a rational number.

 

Example 3: Consider a number 12/-32. Now, let’s see whether it is a fraction or rational number.

Solution: In the number 12/-32, the denominator is negative, i.e., it is not a natural number and a number is said to be a fraction if its denominator is a natural number.

Hence, it is clear that the number 12/-32 is a rational number but not a fraction.    

 

Example 4: By using number 4 and 2/-3; show the difference between a fraction and a rational number.

Solution: i) The number 4 can be expressed as 8/2, i.e., p/q

Where, 8 = p ( numerator) and 2 = q (denominator).

Hence, in this form, i.e., p/q (8/2), 4 is a fraction as well as a rational number.

But, the number 4 in itself is not a fraction as it cannot be represented in p/q form.     

ii) 2/-3

The number 2/-3 is a rational number as the numerator and denominator of a rational number can be negative. But, it is not a fraction because a fraction is always positive.  

 

Example 5: How a fraction is different from a rational number? Show with the help of an example.

Solution: Fraction refers to a part of a whole number whereas a rational number may or may not be a part of any whole number. For instance, 2/2 is undoubtedly a rational number but not a fraction.