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.