Find three numbers in the form \( \frac{p}{q} \) (where \( q \neq 0 \)) between the rational numbers 5 and 9. Also, what property must the rational number \(\frac{p}{q} \) have for... Find three numbers in the form \( \frac{p}{q} \) (where \( q \neq 0 \)) between the rational numbers 5 and 9. Also, what property must the rational number \(\frac{p}{q} \) have for its decimal expansion to be terminating and what property must it have for it to be non-terminating? Read more

Steps to Solve

1. Identify the range We need to find three rational numbers between 5 and 9.

2. Choose rational numbers A rational number can be expressed in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.

3. Select numbers directly between 5 and 9 Here are three rational numbers:

• $\frac{11}{2} = 5.5$
• $\frac{13}{3} \approx 4.33$
• $\frac{17}{4} = 4.25$

4. Confirm they are within the range

• $5 < 5.5 < 9$
• $5 < 4.33 < 9$
• $5 < 4.25 < 9$

5. Discuss terminating decimal property A rational number has a terminating decimal expansion if its denominator (in simplest form) has only the prime factors 2 and/or 5.

6. Explain non-terminating decimal property If a rational number has a denominator with prime factors other than 2 and 5, its decimal expansion will be non-terminating and non-recurring.