Rational Number Between Two Rational Numbers

A rational number is a number which can be written in the form of \(p/q\) (ratio) where the denominator(q) is not equal to zero. This implies that it can be represented in the form of a fraction. Thus, every rational number has a numerator and a denominator, that is, one integer divided by another integer (excluding zero).

Rational Number

Let us first go through some of the examples and then learn how to find rational numbers between two other rational numbers.

Finding Rational Numbers between Two Rational Number

  • The simplest method to find a rational number between 2 rational numbers x and y is to divide their sum by 2.

Example: \(\frac{3}{1}~and~\frac{4}{1}\). The rational number between them is (3 + 4)/2 = \(\frac{3+4}{2}=\frac{7}{2} \)

Example: For numbers with common denominator like \(\frac{3}{20}~and~\frac{9}{20}\) , the rational number between them can be \(\frac{4}{20},~ \frac{5}{20},~\frac{6}{20},~\frac{7}{20},~\frac{8}{20}\), etc.

  • For a negative and a positive number like \(\frac{3}{-3}~and~\frac{3}{20}\) , we have \(\frac{-2}{20},~ \frac{-1}{20},~\frac{0}{20},~\frac{1}{20},~\frac{2}{20}\), etc.
  • We can write \(\frac{3}{20}\) as \(\frac{30}{200}\) and \(\frac{9}{20}\) as \(\frac{90}{200}\) . Thus, rational number between \(\frac{3}{20}~and~\frac{9}{20}\) can also be \(\frac{31}{200},~ \frac{32}{200},~\frac{33}{200},~\frac{34}{200},……..~\frac{89}{200}\) and so on .
  • For finding rational numbers between two rational numbers with different denominators, we first find their equivalent fraction with the same denominator and then find the rational number between them. For example, for finding numbers between \(\frac{-1}{3}~and~\frac{5}{9}\). , we convert \(\frac{-1}{3}~to~\frac{-1\times 3}{3 \times 3}=\frac{-3}{9}\). Now, we have  \(\frac{-2}{9},~ \frac{-1}{9},~\frac{0}{9},~\frac{1}{9},~\frac{2}{9},~\frac{3}{9},~\frac{4}{9}\) between them.

Rational number form one of the major part of mathematics and one cannot deny the importance of mathematics in today’s world. Mathematics has become omnipresent, whether counting numbers at a grocery shop to pay money or making rockets or the unfolding patterns of the universe, math finds its presence in almost all aspects of life and even beyond. Learning math unfolds many things that science could not.

Practise This Question

Which of the following represents a set of 4 rational numbers between 12 and 13 ?