The question is asking for a rational number that lies between two given rational numbers, $\frac{p}{q}$ and $\frac{r}{s}$.

The general formula for a rational number that lies between two rational numbers is given by the arithmetic mean of the two numbers. The arithmetic mean ensures that the resultant value lies between the two rational numbers.

The arithmetic mean of $\frac{p}{q}$ and $\frac{r}{s}$ is:

$\text{Arithmetic Mean} = \frac{1}{2} \left(\frac{p}{q} + \frac{r}{s}\right)$

This corresponds to Option (3): $\frac{1}{2} \left[ \frac{p}{q} + \frac{r}{s} \right]$.

Let me know if you need a detailed explanation or examples to further clarify! Here are some related questions for deeper understanding:

1. Why is the arithmetic mean always between two numbers?
2. What happens if $p/q$ and $r/s$ are equal? Does the mean change?
3. Can you find other rational numbers (besides the mean) between $p/q$ and $r/s$?
4. How would the calculation change if $q$ or $s$ were negative?
5. Is the mean formula valid for irrational numbers as well?

Tip: Always simplify fractions before calculating the arithmetic mean to make computations easier.