https://file.aiquickdraw.com/mathbot/file/1727306082475fno8lq8a.jpg

The problem in the image involves comparing three rational numbers: \( \frac{p}{q}, \frac{r}{s}, \) and \( \frac{t}{u} \). The relationships given are:

1. \( \frac{p}{q} \) is greater than \( \frac{r}{s} \).
2. \( \frac{r}{s} \) is greater than \( \frac{t}{u} \).

You are asked to compare \( \frac{p}{q} \) and \( \frac{t}{u} \), and justify your reasoning on a number line.

### Steps to Solve:
- Since \( \frac{p}{q} > \frac{r}{s} \) and \( \frac{r}{s} > \frac{t}{u} \), it follows that \( \frac{p}{q} > \frac{t}{u} \) by transitive property (if \( a > b \) and \( b > c \), then \( a > c \)).
  
- On a number line, when \( \frac{p}{q} \) is greater than \( \frac{r}{s} \), \( \frac{p}{q} \) would be located to the right of \( \frac{r}{s} \). Similarly, since \( \frac{r}{s} \) is greater than \( \frac{t}{u} \), \( \frac{r}{s} \) would be to the right of \( \frac{t}{u} \).

- Therefore, \( \frac{p}{q} \) is to the right of \( \frac{t}{u} \) on the number line.

### Filling in the Blanks:
- The first blank asks where \( \frac{p}{q} \) is relative to \( \frac{r}{s} \). The answer is: **to the right**.
- The second blank asks where \( \frac{r}{s} \) is relative to \( \frac{t}{u} \). The answer is: **to the right**.
- The third blank asks where \( \frac{p}{q} \) is relative to \( \frac{t}{u} \). The answer is: **to the right**.

Would you like more details or explanations?

Here are 5 related questions to expand on this concept:
1. What is the transitive property of inequalities?
2. How do you compare two fractions with different denominators?
3. What is the significance of ordering numbers on a number line?
4. Can you give an example where three fractions are ordered differently?
5. What methods can you use to visualize inequalities beyond the number line?

**Tip:** When comparing fractions, converting them to have a common denominator can make the comparison much easier.

Suppose p/q, r/s, and t/u represent three rational numbers. If p/q is greater than r/s and r/s is greater than t/u, compare p/q and t/u. Explain your reasoning.

Learn how to compare rational numbers using the transitive property of inequalities. This math problem involves understanding how to compare fractions and their positions on a number line. Suitable for students in Grades 6-8.

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