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?

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Understand the Problem

The question is asking to find three rational numbers between two given rational numbers, 5 and 9. It also requests an explanation of the properties that a rational number must satisfy for its decimal expansion to be terminating, as well as clarification on properties related to non-terminating decimal expansions.

Answer

The rational numbers are $\frac{11}{2}, \frac{14}{3}, \frac{17}{4}$.
Answer for screen readers

The three rational numbers found between 5 and 9 are $\frac{11}{2}, \frac{14}{3}, \frac{17}{4}$.

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:

  • $\frac{11}{2} = 5.5$
  • $\frac{13}{3} \approx 4.33$
  • $\frac{17}{4} = 4.25$
  1. Confirm they are within the range
  • $5 < 5.5 < 9$
  • $5 < 4.33 < 9$
  • $5 < 4.25 < 9$
  1. 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.

  2. 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.

The three rational numbers found between 5 and 9 are $\frac{11}{2}, \frac{14}{3}, \frac{17}{4}$.

More Information

Rational numbers are fundamental in number theory and have various applications in real-world scenarios like measurements and financial calculations.

Tips

  • Confusing rational numbers with irrational numbers; remember that rational numbers can be represented as fractions.
  • Miscalculating the range; ensure selected numbers are indeed between 5 and 9.
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