The question asks which number is irrational. Let's analyze each option:

1. $\frac{9}{31}$: This is a fraction of two integers, so it is a rational number.

2. $\sqrt{\frac{9}{31}}$: The square root of a fraction that does not simplify into a perfect square will be irrational. Since 31 is not a perfect square, this value is irrational.

3. $0.\overline{6}$: This is a repeating decimal, which is always rational because it can be expressed as a fraction.

4. $\sqrt{\frac{9}{16}}$: The square root of this fraction simplifies to $\frac{3}{4}$, which is a rational number.

The correct answer is $\sqrt{\frac{9}{31}}$ because it is irrational.

Would you like any further details or clarifications?

Here are 5 related questions:

1. What is the difference between rational and irrational numbers?
2. How do you simplify square roots of fractions?
3. Can all repeating decimals be expressed as fractions?
4. What are examples of perfect square numbers?
5. How can you identify if a square root will result in an irrational number?

Tip: An irrational number is one that cannot be expressed as a simple fraction, often resulting from the square root of a non-perfect square.