The question states that $q$ and $r$ are both irrational numbers, and it asks which statement about $\frac{q}{r}$ is true. Let's analyze the options carefully:

1. $\frac{q}{r}$ is rational.

   • This is not necessarily true because the division of two irrational numbers does not always result in a rational number.

2. $\frac{q}{r}$ is irrational.

   • This is also not always true. While $q$ and $r$ are both irrational, their ratio $\frac{q}{r}$ can sometimes be rational. For example, if $q = r\sqrt{2}$, then $\frac{q}{r} = \sqrt{2}$, which is irrational. However, in other cases, it can simplify to a rational number.

3. $\frac{q}{r}$ can be rational or irrational, depending on the values of $q$ and $r$.

   • This is the correct answer. The ratio $\frac{q}{r}$ can be either rational or irrational, depending on the specific values of $q$ and $r$.

Final Answer:

The correct statement is:
$\frac{q}{r}$ can be rational or irrational, depending on the values of $q$ and $r$.

Do you want a further explanation on irrational numbers or examples? Let me know!

Here are 5 related questions to explore the topic further:

1. What is the definition of an irrational number?
2. Can the product of two irrational numbers ever be rational?
3. Give an example where the ratio of two irrational numbers is rational.
4. Why is $\sqrt{2}$ considered an irrational number?
5. Can the sum of two irrational numbers be rational?

Tip: A number is irrational if it cannot be expressed as a ratio of two integers.