 # Rational Numbers in Standard Form

Expressing a rational number into standard form means there is no common factor, other than 1, available in its numerator and denominator and its denominator is a positive integer.  Numbers which can be expressed in the form of p/q, where p and q are integers and q is not equal to zero, are known as rational numbers.  Hence, if 4/6 is a rational number, then its standard form will be 2/3 since we cannot solve 2/3, any more.

The standard form of rational number helps us to determine the value in a more specific way. Like, 20/25 can be expressed as 4/5, 10/20 can be expressed as 1/2 and so on.

In Maths, we rationalise the fractions to express them into standard forms. The fractions have numerator and denominator part. A fraction is in standard form when numerator and denominator are co-prime. All integers and fractions are rational numbers. It is easy to perform addition and subtraction of rational numbers once we have rationalised their denominators.

## How to Convert Rational Numbers in Standard Form?

A rational number is made up of a numerator and a denominator. A rational number is said to be in standard form if the Highest Common Factor or the H.C.F. of numerator and denominator is 1. Now, the question arises that how to change any rational number into its standard form? ### Procedure to Convert Rational Numbers to Standard Form

1. Whenever we have a rational number, first, we find the H.C.F. of numerator and denominator, if it is 1 i.e. if the numerator and denominator of the rational number are coprime numbers, then the given rational number is in its standard form.
2. If the numerator and denominator are not co-prime, then we start dividing both the numerator and denominator by the common factor of both. We keep on dividing the numerator and denominator with the common factors unless we get a numerator and denominator with H.C.F. equal to 1.

### Examples

Let us consider the following example, to have a better understanding.

Consider a rational number, 16/24. The H.C.F. of 16 and 24 is 8, which is not equal to 1, hence the given rational number is not in its standard form. Now, we know that 2 is a common factor of 16 and 24.  Dividing both the numerator and denominator by 2, we get 8/12.

Again the H.C.F. is not equal to 1, so again dividing it by 2. On dividing the numerator and denominator by 2 we get 4/6. Still, we find that the H.C.F. is not equal to 1. So, we will again divide both the numerator and denominator by 2. So, now we finally obtain 2/3. The H.C.F. of 2 and 3 is 1, i.e. 2 and 3 are co-prime. Hence, the rational numbers obtained now has an H.C.F. equal to 1.

Therefore, the standard form of 16/24 = 2/3

This is the way in which we find how to change any rational numbers in standard form.

## Solved Problems

Question: Write 23/69 in standard form.

Solution: Given, 23/69

If we take out the H.C.F. of 23 and 69, we get;

23 = 1 x 23

69 = 1 x 3 x 23

As we can see here, there is one more factor common between 23 and 69 other than 1, which is 23 itself, therefore if we cancel the common factor from numerator and denominator, we get;

23/69 = 1/3

Question: Which of the following rational numbers is equivalent to 2/3?

(a) 3/2

(b) 4/9

(c) 4/6

(d) 9/4

4 = 2 x 2

6 = 2 x 3

Since 2 is the common factor here, then we will cancel it from numerator and denominator.

Therefore, 4/6 = 2/3