Can you think of other way(s) to find a rational number between any two rational numbers?

Yes, there are several ways to find a rational number between any two rational numbers a and b :

1. Average (Mean) method : The average of a and b, given by $\dfrac{a + b}{2}$, always lies between a and b.

For example, between $\dfrac{1}{2}$ and $\dfrac{3}{4}$ :

Average =

$= \dfrac{\dfrac{1}{2} + \dfrac{3}{4}}{2} \\[1em] = \dfrac{\dfrac{5}{4}}{2} \\[1em] = \dfrac{5}{4} \times \dfrac{1}{2} \\[1em] = \dfrac{5}{8}$

2. Common denominator method : Convert both rational numbers to a common denominator and select numerators in between. If the gap is too small, multiply numerator and denominator by a larger number to increase the gap.

For example, between $\dfrac{1}{3}$ and $\dfrac{1}{2}$ :

⇒ $\dfrac{1}{3} = \dfrac{2}{6}$, $\dfrac{1}{2} = \dfrac{3}{6}$. To find more numbers, write as $\dfrac{20}{60}$ and $\dfrac{30}{60}$. Then $\dfrac{21}{60}, \dfrac{22}{60}, \ldots, \dfrac{29}{60}$ all lie between them.

3. Decimal method : Convert both rational numbers into decimals and pick decimal numbers between them by extending the decimal places.

4. Weighted average method : Use $\dfrac{2a + b}{3}$ or $\dfrac{a + 2b}{3}$, which divide the interval into thirds.

Hence, there are multiple ways to find a rational number between any two rational numbers — the average method, common denominator method, decimal method and weighted averages.