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

Question

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Answer 1

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 \( \frac{5}{12} \).