Which one of the following rational numbers has non-terminating and repeating decimal expression?
17/6
Understanding Rational Numbers and Decimal Expansions
Rational numbers are numbers that can be expressed in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). When we divide the numerator by the denominator, we get a decimal expansion. This decimal expansion can be either terminating or non-terminating and repeating.
How to Identify Terminating vs. Non-terminating Decimal Expansions
A key property helps us determine the type of decimal expansion without actually performing the division. A rational number \(\frac{p}{q}\), where \(p\) and \(q\) are coprime integers (meaning they have no common factors other than 1), has:
- A terminating decimal expansion if the prime factorization of the denominator, \(q\), contains only the prime numbers 2 and/or 5.
- A non-terminating and repeating decimal expansion if the prime factorization of the denominator, \(q\), contains any prime factor other than 2 or 5.
First, we must ensure the rational number is in its simplest form before analyzing the denominator.
Analyzing Each Rational Number Option
Let's examine each given rational number to determine the nature of its decimal expansion.
Option 1: \(\frac{15}{1600}\)
- Simplify the fraction: \(\frac{15}{1600} = \frac{3 \times 5}{320 \times 5} = \frac{3}{320}\). The fraction is in simplest form.
- Find the prime factorization of the denominator, 320: \(320 = 32 \times 10 = 2^5 \times 2 \times 5 = 2^6 \times 5\).
- The prime factors of the denominator are only 2 and 5.
- Conclusion: This rational number has a terminating decimal expansion.
Option 2: \(\frac{23}{8}\)
- Simplify the fraction: The fraction \(\frac{23}{8}\) is already in simplest form because 23 is a prime number and 8 is \(2^3\). They have no common factors.
- Find the prime factorization of the denominator, 8: \(8 = 2^3\).
- The only prime factor of the denominator is 2.
- Conclusion: This rational number has a terminating decimal expansion.
Option 3: \(\frac{35}{50}\)
- Simplify the fraction: \(\frac{35}{50} = \frac{5 \times 7}{5 \times 10} = \frac{7}{10}\). The fraction is in simplest form.
- Find the prime factorization of the denominator, 10: \(10 = 2 \times 5\).
- The prime factors of the denominator are only 2 and 5.
- Conclusion: This rational number has a terminating decimal expansion.
Option 4: \(\frac{17}{6}\)
- Simplify the fraction: The fraction \(\frac{17}{6}\) is already in simplest form because 17 is a prime number and 6 is \(2 \times 3\). They have no common factors.
- Find the prime factorization of the denominator, 6: \(6 = 2 \times 3\).
- The prime factors of the denominator are 2 and 3. Since the prime factor 3 is present, which is not 2 or 5, the decimal expansion will be non-terminating and repeating.
- Conclusion: This rational number has a non-terminating and repeating decimal expansion.
Summary of Decimal Expansions
| Rational Number | Simplified Form | Denominator Prime Factors | Decimal Expansion Type |
|---|---|---|---|
| \(\frac{15}{1600}\) | \(\frac{3}{320}\) | \(2^6 \times 5\) (only 2 and 5) | Terminating |
| \(\frac{23}{8}\) | \(\frac{23}{8}\) | \(2^3\) (only 2) | Terminating |
| \(\frac{35}{50}\) | \(\frac{7}{10}\) | \(2 \times 5\) (only 2 and 5) | Terminating |
| \(\frac{17}{6}\) | \(\frac{17}{6}\) | \(2 \times 3\) (includes 3) | Non-terminating and Repeating |
Based on the analysis, the rational number \(\frac{17}{6}\) has a non-terminating and repeating decimal expression.
Revision Table: Rational Numbers and Decimal Types
| Denominator Prime Factors (in simplest form) | Decimal Expansion Type |
|---|---|
| Only 2s and/or 5s | Terminating |
| Contains factors other than 2 or 5 | Non-terminating and Repeating |
Additional Information: Examples of Decimal Expansions
- \(\frac{1}{2} = 0.5\) (Terminating) - Denominator is 2.
- \(\frac{1}{4} = 0.25\) (Terminating) - Denominator is \(2^2\).
- \(\frac{1}{5} = 0.2\) (Terminating) - Denominator is 5.
- \(\frac{1}{10} = 0.1\) (Terminating) - Denominator is \(2 \times 5\).
- \(\frac{1}{3} = 0.333... = 0.\bar{3}\) (Non-terminating Repeating) - Denominator is 3.
- \(\frac{1}{6} = 0.1666... = 0.1\bar{6}\) (Non-terminating Repeating) - Denominator is \(2 \times 3\).
- \(\frac{1}{7} = 0.142857142857... = 0.\overline{142857}\) (Non-terminating Repeating) - Denominator is 7.
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