## Introduction to 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.

Fractions are used by us in our daily lives all the time and we don’t even realize it. Hence, it is important to learn about them not only for your Math lessons and exams but also, for daily calculations as a life skill to be used throughout your lifetime.

Fractions form a part of our mental Math in the smallest instances around us. When you go shopping with your family, there are festive offers in shops saying 'Buy now, Pay Half'. Remember, a fraction represents a part of a whole. Similarly, the mind sees the discount 'half' of the initial, undiscounted price. When you share the cake with your friends, you cut the cake into equal parts or 'fractions' equal to the number of people in the group.

You may not feel it, but fractions are everywhere. Even when someone asks you the time. The clock says 'quarter to 12'. It means 15 minutes remaining until 12 O’clock. Fractions are used in recipes that tell you to pour '1/4th of the bottle' and ' 1/2 a spoon of honey real-life. A lot of competitive sports like F1 motorsport, NASCAR, athletics and cricket use fractions to monitor reaction times, gaps like 'one-tenth of a second' between their competitors and targets to chase like fastest lap times and fastest pit stops.

Even when you go to the supermarket to get groceries, you buy 'half a dozen bananas' which is nothing but 1/6 of a dozen. When you have to compare things in terms of height, weight and mass, fractions are used to say things like 'I used to be at least half of how fat I am right now' and 'how can one sibling be 1/3rd the height of the other one?'.

Now that you know how integral fractions are in everyday life, you can learn all about them and real life calculation based problems related to fractions on Vedantu.

### 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:

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.

## FAQs on Difference Between Fraction and Rational Numbers

**1. Why can a fraction not have zero as a denominator?**

A fraction cannot have zero as a denominator because division by zero means that all values will be undefined. Undefined values have no meaning. Hence, it is important to remember that the expression is not a legal fraction if the denominator is a zero.

**2. Can a fraction be greater than 1?**

Yes, a fraction can be greater than 1. A thumb rule to remember if a fraction is greater than 1 is to see that the numerator is always greater than the denominator. Similarly, a fraction equals less than 1 when the numerator is always lesser than the denominator.

**3. Can a fraction have zero as a numerator?**

Yes, a fraction can have zero as a numerator. An expression is considered to be a legal fraction even if it has zero as its numerator because it doesn't lead to all values as undefined. It leads to the entire fraction's value being equal to zero. For example, 0/23 is equal to 0, 0/50 also equals zero. In fact, The value of all fractions with zero in the numerator equals zero.

**4. Is it true that humans sleep away 1/3rd of their life?**

On average, studies suggest that humans require 8 hours of sleep in a day to lead a healthy lifestyle. A day has 24 hours in total. So, this means that humans sleep 8 hours out of 24 hours, which can be represented as a fraction, 8/24. This can be simplified to 1/3. This means 1/3rd of a day is 8 hours. Hence, if a human sleeps 8 hours on all days, he or she sleeps 1/3rd of all his or her life.

**5. How to convert a mixed fraction to an improper fraction?**

In order to convert a mixed fraction into an improper fraction, multiply the whole number with the denominator of the proper fraction. The product of this multiplication should be added to the numerator of the proper fraction. This result of the addition is the numerator of the improper fraction. The denominator of the improper fraction will remain the same as the denominator of the proper fraction.

To see examples of this, students can learn from Vedantu where all topics regarding addition, subtraction, multiplication and division of fractions and Rational Numbers are covered. Students can download study material for free from the Vedantu website and app.