Question
Marais rewrites a fraction less than 1 as a decimal the numerator is a whole number grater than 0 which denominator will the fraction always convert to repeating decimals?
6
15
11
18
6
15
11
18
Answers
GPT-4o mini
To determine which denominator will always convert a fraction less than 1 into a repeating decimal, we need to consider how decimal representation works with rational fractions.
A fraction will convert to a repeating decimal if the denominator (after simplifying the fraction) has prime factors other than 2 and 5. If the fraction's denominator (after simplification) has only the prime factors 2 and/or 5, then it will convert to a terminating decimal.
Let's look at the denominators you provided:
- **6**: \(6 = 2 \times 3\) (has a factor of 3, so it will be a repeating decimal)
- **15**: \(15 = 3 \times 5\) (has a factor of 3, so it will be a repeating decimal)
- **11**: \(11\) is prime and not 2 or 5 (so, it will be a repeating decimal)
- **18**: \(18 = 2 \times 3^2\) (has a factor of 3, so it will be a repeating decimal)
Since all options (6, 15, 11, and 18) contain a factor that leads to a repeating decimal, we can summarize:
- **All the denominators given will lead to repeating decimals when used in a fraction less than 1**, but the question asks which will **always** convert. Here, the unique prime factor, which is **11**, will always convert to a repeating decimal when used as the denominator, providing the clearest case of the property.
Therefore, **the answer is 11.**
A fraction will convert to a repeating decimal if the denominator (after simplifying the fraction) has prime factors other than 2 and 5. If the fraction's denominator (after simplification) has only the prime factors 2 and/or 5, then it will convert to a terminating decimal.
Let's look at the denominators you provided:
- **6**: \(6 = 2 \times 3\) (has a factor of 3, so it will be a repeating decimal)
- **15**: \(15 = 3 \times 5\) (has a factor of 3, so it will be a repeating decimal)
- **11**: \(11\) is prime and not 2 or 5 (so, it will be a repeating decimal)
- **18**: \(18 = 2 \times 3^2\) (has a factor of 3, so it will be a repeating decimal)
Since all options (6, 15, 11, and 18) contain a factor that leads to a repeating decimal, we can summarize:
- **All the denominators given will lead to repeating decimals when used in a fraction less than 1**, but the question asks which will **always** convert. Here, the unique prime factor, which is **11**, will always convert to a repeating decimal when used as the denominator, providing the clearest case of the property.
Therefore, **the answer is 11.**