Asked by I_EAT_EVERYDAY
Which of the following rational numbers can be expressed as a repeating decimal: 7 over 8, 5 over 12, 17 over 25, or 31 over 40
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Answers
Answered by
GPT-4o mini
A rational number can be expressed as a repeating decimal if, when simplified to its lowest terms, its denominator has prime factors other than 2 and 5. To determine if a given rational number can be expressed as a repeating decimal, we need to factor the denominators.
1. \( \frac{7}{8} \)
- The denominator \( 8 = 2^3 \). Since there are no factors other than 2, this is a terminating decimal.
2. \( \frac{5}{12} \)
- The denominator \( 12 = 2^2 \times 3 \). Since there is a prime factor of 3, this number can be expressed as a repeating decimal.
3. \( \frac{17}{25} \)
- The denominator \( 25 = 5^2 \). Since there are no factors other than 5, this is a terminating decimal.
4. \( \frac{31}{40} \)
- The denominator \( 40 = 2^3 \times 5 \). Since there are no prime factors other than 2 and 5, this is a terminating decimal.
Based on this analysis, the only rational number from the provided options that can be expressed as a repeating decimal is \( \frac{5}{12} \).
1. \( \frac{7}{8} \)
- The denominator \( 8 = 2^3 \). Since there are no factors other than 2, this is a terminating decimal.
2. \( \frac{5}{12} \)
- The denominator \( 12 = 2^2 \times 3 \). Since there is a prime factor of 3, this number can be expressed as a repeating decimal.
3. \( \frac{17}{25} \)
- The denominator \( 25 = 5^2 \). Since there are no factors other than 5, this is a terminating decimal.
4. \( \frac{31}{40} \)
- The denominator \( 40 = 2^3 \times 5 \). Since there are no prime factors other than 2 and 5, this is a terminating decimal.
Based on this analysis, the only rational number from the provided options that can be expressed as a repeating decimal is \( \frac{5}{12} \).