Let's analyze the question:

"Which number set contains all rational numbers?"

To solve this, we need to identify which set contains only rational numbers. A rational number is any number that can be expressed as the ratio of two integers (i.e., in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$).

Let's analyze each set:

1. Set 1: $\left\{\frac{1}{3}, -3.45, \sqrt{9}\right\}$

   • $\frac{1}{3}$ is rational.
   • $-3.45$ is rational (since it can be written as $-\frac{345}{100}$).
   • $\sqrt{9} = 3$, which is also a rational number.

   Thus, all numbers in this set are rational.

2. Set 2: $\left\{\pi, 9.25, \sqrt{37}\right\}$

   • $\pi$ is irrational.
   • $9.25$ is rational (can be written as $\frac{925}{100}$).
   • $\sqrt{37}$ is irrational because 37 is not a perfect square.

   Thus, this set contains irrational numbers.

3. Set 3: $\left\{\sqrt{2}, 10, 7\right\}$

   • $\sqrt{2}$ is irrational.
   • $10$ and $7$ are rational.

   Thus, this set contains irrational numbers.

4. Set 4: $\left\{3, -9, \sqrt{44}\right\}$

   • $3$ and $-9$ are rational.
   • $\sqrt{44}$ is irrational because 44 is not a perfect square.

   Thus, this set contains irrational numbers.

Conclusion:

The first set $\left\{\frac{1}{3}, -3.45, \sqrt{9}\right\}$ contains only rational numbers. Hence, the correct answer is Set 1.

Do you want further clarification on any of these points?

Here are 5 related questions for further practice:

1. What is the definition of an irrational number?
2. Can the square root of any perfect square be a rational number?
3. How can we convert repeating decimals into fractions (rational numbers)?
4. Is $\sqrt{2}$ a rational or an irrational number? Why?
5. How can you determine if a number expressed as a square root is rational?

Tip: Always remember, if a square root simplifies to an integer, the number is rational!