The problem states that Megan added two numbers, and the resulting sum was an irrational number. The first number was $\frac{2}{3}$, and we need to determine which of the following could be the second number to make the sum irrational.

Key Points:

• The first number, $\frac{2}{3}$, is a rational number.
• The sum of two rational numbers is always rational. Therefore, to obtain an irrational sum, the second number must be irrational.
• We need to identify the irrational number from the choices provided.

Let's analyze the options:

• Option A: $\frac{6}{5}$
  $\frac{6}{5}$ is a rational number (since it's a ratio of two integers). Therefore, adding this to $\frac{2}{3}$ will result in a rational number.

• Option B: $3\pi$
  $\pi$ is irrational, and multiplying it by 3 keeps it irrational. Adding $3\pi$ to $\frac{2}{3}$ results in an irrational number, since the sum of a rational and an irrational number is always irrational. Thus, this could be the correct answer.

• Option C: 5
  5 is a rational number. Adding this to $\frac{2}{3}$ will give a rational number, not an irrational one.

• Option D: $\sqrt{9}$
  $\sqrt{9} = 3$, which is a rational number. Adding 3 to $\frac{2}{3}$ results in a rational sum.

Conclusion:

The second number that could lead to an irrational sum is Option B: $3\pi$.

Would you like further clarification or details on irrational numbers? Here are 5 related questions to explore:

1. What is the difference between rational and irrational numbers?
2. How can the sum of a rational and an irrational number be irrational?
3. Can the product of a rational and an irrational number ever be rational?
4. How can you prove that $\pi$ is irrational?
5. What are some examples of irrational numbers besides $\pi$?

Tip: Whenever you're asked to determine if a number is irrational, check if it can be written as a fraction of two integers—if not, it's irrational!