The average of two rational numbers $a$ and $b$ is $\frac{a+b}{2}$. Since $a$ and $b$ are rational, their sum $a+b$ is also a rational number by the rules of addition of rational numbers. Dividing this sum by 2 (a non-zero rational number) using the rules of division yields another rational number, $\frac{a+b}{2}$.

Furthermore, this average always lies between $a$ and $b$. If $a < b$, adding $a$ to both sides gives $2a < a+b$, which simplifies to $a < \frac{a+b}{2}$. Similarly, adding $b$ to both sides of $a < b$ gives $a+b < 2b$, which simplifies to $\frac{a+b}{2} < b$.

Therefore, $\frac{a+b}{2}$ is always a rational number strictly between $a$ and $b$, illustrating the density of rational numbers.