### 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 $\frac{-3}{2}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\frac{-2}{3}$.

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 $$6$$.

$\frac{-3×3}{2×3}=\frac{-9}{6}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\frac{-2×2}{3×2}=\frac{-4}{6}$

We have,

$\frac{-9}{6}<\frac{-8}{6}<\frac{-7}{6}<\frac{-6}{6}<\frac{-5}{6}<\frac{-4}{6}$

(or)

$\frac{-3}{2}<\frac{-8}{6}<\frac{-7}{6}<\frac{-6}{6}<\frac{-5}{6}<\frac{-2}{3}$

She could find rational numbers $\frac{-8}{6},\frac{-7}{6},\frac{-6}{6},\frac{-5}{6}$ between $\frac{-3}{2}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\frac{-2}{3}$.

She doubts that are there only $$4$$ rational numbers between $\frac{-3}{2}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\frac{-2}{3}$. She gets to know there are more than $$4$$ rational numbers between $\frac{-3}{2}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\frac{-2}{3}$ because if we find the multiples of denominators, then many more rational numbers can be inserted between $\frac{-3}{2}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\frac{-2}{3}$.

For example,

$\frac{-3}{2}=\frac{-3×4}{2×4}=\frac{-12}{8}=\frac{-3×10}{2×10}=\frac{-30}{20}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\frac{-2}{3}=\frac{-2×4}{3×4}=\frac{-8}{12}=\frac{-2×10}{3×10}=\frac{-20}{30}$

Now, between $\frac{-30}{20}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\frac{-20}{30}$, there are $$9$$ rational numbers present.