Asked by TheOGBlurryface
Which of the following rational numbers can be expressed as a repeating decimal: 7/8
, 5/12
, 17/25
, or 31/40
?
, 5/12
, 17/25
, or 31/40
?
Answers
Answered by
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 the fractions when they are in their simplest form.
A rational number can be expressed as a terminating decimal if the only prime factors of its denominator (after simplification) are 2 and/or 5. If there are any other prime factors, it will be a repeating decimal.
Let's analyze each of the given fractions:
1. **7/8**: The denominator is 8, which is \(2^3\). Since it consists only of the prime factor 2, \(7/8\) is a terminating decimal.
2. **5/12**: The denominator is 12, which factors to \(2^2 \times 3\). The presence of the factor 3 means that \(5/12\) is a repeating decimal.
3. **17/25**: The denominator is 25, which is \(5^2\). Since it consists only of the prime factor 5, \(17/25\) is a terminating decimal.
4. **31/40**: The denominator is 40, which factors to \(2^3 \times 5\). Since it consists only of the prime factors 2 and 5, \(31/40\) is a terminating decimal.
In conclusion, the only rational number from the list that can be expressed as a repeating decimal is **5/12**.
A rational number can be expressed as a terminating decimal if the only prime factors of its denominator (after simplification) are 2 and/or 5. If there are any other prime factors, it will be a repeating decimal.
Let's analyze each of the given fractions:
1. **7/8**: The denominator is 8, which is \(2^3\). Since it consists only of the prime factor 2, \(7/8\) is a terminating decimal.
2. **5/12**: The denominator is 12, which factors to \(2^2 \times 3\). The presence of the factor 3 means that \(5/12\) is a repeating decimal.
3. **17/25**: The denominator is 25, which is \(5^2\). Since it consists only of the prime factor 5, \(17/25\) is a terminating decimal.
4. **31/40**: The denominator is 40, which factors to \(2^3 \times 5\). Since it consists only of the prime factors 2 and 5, \(31/40\) is a terminating decimal.
In conclusion, the only rational number from the list that can be expressed as a repeating decimal is **5/12**.