Question

How many rational numbers exist between any two distinct rational numbers.
\[
  A.{\text{ 2}} \\
  B.{\text{ 3}} \\
  C.{\text{ 11}} \\
  D.{\text{ Infinite number of rational numbers}}{\text{.}} \\
 \]

Answer 1

Hint: In order to solve such a problem take two rational numbers and find some rational number in between them in order to develop a general idea. Also the problem can be solved by a general definition of rational numbers.

Complete step-by-step answer:
Let us solve the problem practically by taking two rational numbers, say \[a{\text{ and }}b\] .
Now taking the mean of \[a{\text{ and }}b\] , we get
Mean = $\dfrac{{a + b}}{2}$ , which is in between \[a{\text{ and }}b\] and is also a rational number.
Similarly if we take the mean of $\dfrac{{a + b}}{2}{\text{ and }}b$ , we get another rational number and so on.
So from the above process we find that every time we take the mean of 2 different numbers within two numbers, we get another rational number.
Hence, there exists an infinite number of rational numbers within two rational numbers.
So, option D is the correct option.

Note: A rational number is a number that can be expressed as the quotient or fraction $\dfrac{p}{q}$ of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. But every rational number is not an integer.