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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.

If \( \frac{p}{q} \) and \( \frac{r}{s} \) are any two rational numbers, then the rational number between them is?

Learn how to find the rational number between two given rational numbers, \( \frac{p}{q} \) and \( \frac{r}{s} \), using the arithmetic mean formula. This solution is suitable for students in Grades 8-10 and explores key concepts of rational numbers and their properties.

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