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Which of the following rational numbers can be expressed as a repeating decimal: 7/8 , 5/12 , 17/25 , or 31/40 ?Asked by Lee Hooni 🖤
Which of the following rational numbers can be expressed as a repeating decimal: 78
, 5/12
, 17/25
, or 31/40
?
, 5/12
, 17/25
, or 31/40
?
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Answered by
GPT-4o mini
A rational number can be expressed as a repeating decimal if its denominator (when in simplest form) has prime factors other than only 2 or 5. Let's analyze the given fractions:
1. **78**: This is an integer and can be expressed as a decimal (78.0), which does not repeat.
2. **5/12**: The denominator is 12, which factors to \(2^2 \times 3\). Because it has the factor 3 (not just 2 or 5), it will be a repeating decimal.
3. **17/25**: The denominator is 25, which factors to \(5^2\). Since it only contains the factor 5, this will be a terminating decimal.
4. **31/40**: The denominator is 40, which factors to \(2^3 \times 5\). Since it only contains the factors 2 and 5, this will also be a terminating decimal.
Based on this analysis, the rational number that can be expressed as a repeating decimal is **5/12**.
1. **78**: This is an integer and can be expressed as a decimal (78.0), which does not repeat.
2. **5/12**: The denominator is 12, which factors to \(2^2 \times 3\). Because it has the factor 3 (not just 2 or 5), it will be a repeating decimal.
3. **17/25**: The denominator is 25, which factors to \(5^2\). Since it only contains the factor 5, this will be a terminating decimal.
4. **31/40**: The denominator is 40, which factors to \(2^3 \times 5\). Since it only contains the factors 2 and 5, this will also be a terminating decimal.
Based on this analysis, the rational number that can be expressed as a repeating decimal is **5/12**.
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