Theory:

Asha wanted to count the numbers between \(-9\) and \(12\). She found that totally \(20\) integers are present, with \(8\) negative integers, \(zero\) and \(11\) positive integers present between \(-9\) and \(12\) (excluding \(-9\) and \(12\)).
 
She also gets to know that there are no integers, present between any two consecutive integers, that is, for example, between \(8\) and \(9\) there is no integer.
 
So, between any two consecutive integers the number of integers is \(0\).
 
She wondered whether the same would happen in case of rational numbers too?
 
She took two rational numbers 32and23.
 
She converted them to rational numbers with same denominators, for which the denominator of both the numbers should be converted into LCM.
 
She found the LCM of (\(2\),\(3\)) \(= 6\).
 
Convert both the numbers with denominator as \(6\).
 
3×32×3=96and2×23×2=46
 
We have,
 
96<86<76<66<56<46
 
(or)
 
32<86<76<66<56<23
 
She could find rational numbers 86,76,66,56 between 32and23.
 
She doubts that are there only \(4\) rational numbers between 32and23. She gets to know there are more than \(4\) rational numbers between 32and23 because if we find the multiples of denominators, then many more rational numbers can be inserted between 32and23.
 
For example,
 
32=3×42×4=128=3×102×10=3020and23=2×43×4=812=2×103×10=2030
 
Now, between 3020and2030, there are \(9\) rational numbers present.