Question
Which of the following rational numbers can be expressed as a repeating decimal out of 7/8 or 5/12 or 17/25 or 31/40
Answers
GPT-4o mini
To determine which of the given rational numbers can be expressed as a repeating decimal, we need to analyze the denominators of each fraction. A rational number will have a terminating decimal if the denominator (when simplified) has only the prime factors of 2 and/or 5. If the denominator has any other prime factors, the decimal will be repeating.
Let's analyze each fraction:
1. **\( \frac{7}{8} \)**:
- Denominator: \( 8 = 2^3 \)
- Only prime factor is 2, so \( \frac{7}{8} \) has a terminating decimal.
2. **\( \frac{5}{12} \)**:
- Denominator: \( 12 = 2^2 \times 3 \)
- Contains a prime factor of 3, so \( \frac{5}{12} \) has a repeating decimal.
3. **\( \frac{17}{25} \)**:
- Denominator: \( 25 = 5^2 \)
- Only prime factor is 5, so \( \frac{17}{25} \) has a terminating decimal.
4. **\( \frac{31}{40} \)**:
- Denominator: \( 40 = 2^3 \times 5 \)
- Only prime factors are 2 and 5, so \( \frac{31}{40} \) has a terminating decimal.
Based on this analysis, the only fraction that can be expressed as a repeating decimal is:
**\( \frac{5}{12} \)**.
Let's analyze each fraction:
1. **\( \frac{7}{8} \)**:
- Denominator: \( 8 = 2^3 \)
- Only prime factor is 2, so \( \frac{7}{8} \) has a terminating decimal.
2. **\( \frac{5}{12} \)**:
- Denominator: \( 12 = 2^2 \times 3 \)
- Contains a prime factor of 3, so \( \frac{5}{12} \) has a repeating decimal.
3. **\( \frac{17}{25} \)**:
- Denominator: \( 25 = 5^2 \)
- Only prime factor is 5, so \( \frac{17}{25} \) has a terminating decimal.
4. **\( \frac{31}{40} \)**:
- Denominator: \( 40 = 2^3 \times 5 \)
- Only prime factors are 2 and 5, so \( \frac{31}{40} \) has a terminating decimal.
Based on this analysis, the only fraction that can be expressed as a repeating decimal is:
**\( \frac{5}{12} \)**.