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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?

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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!

Which number set contains all rational numbers?

Learn how to identify which set contains only rational numbers. This problem involves understanding the properties of rational and irrational numbers, and is suitable for students in Grades 6-8.

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