When converting fractions to decimals, some fractions result in repeating decimals while others do not. To determine which fractions result in repeating decimals, we need to look at the nature of their denominators.
A fraction in its simplest form results in a repeating decimal if its denominator, after factoring out any common factors with the numerator, has prime factors other than 2 or 5. Otherwise, it will terminate. Let's analyze each fraction given:
A. \( \frac{3}{4} \)
- The denominator is 4, which factors into \(2 \times 2\), having only the prime factors 2.
- Therefore, the decimal representation of \( \frac{3}{4} \) is \(0.75\), which is a terminating decimal.
B. \( \frac{1}{9} \)
- The denominator is 9, which factors into \(3 \times 3\), having only the prime factor 3.
- Therefore, the decimal representation of \( \frac{1}{9} \) repeats because 9 is not composed solely of the prime factors 2 or 5. The decimal is \(0.\overline{1}\) (repeating).
C. \( \frac{5}{11} \)
- The denominator is 11, which is a prime number and does not consist exclusively of the prime factors 2 or 5.
- Therefore, the decimal representation of \( \frac{5}{11} \) repeats. The decimal is approximately \(0.\overline{45}\), indicating repetition.
D. \( \frac{3}{7} \)
- The denominator is 7, which is a prime number and does not consist exclusively of the prime factors 2 or 5.
- Therefore, the decimal representation of \( \frac{3}{7} \) repeats. The decimal is approximately \(0.\overline{428571}\), indicating repetition.
Based on the analysis, the fractions that result in repeating decimals are:
B. \( \frac{1}{9} \)
C. \( \frac{5}{11} \)
D. \( \frac{3}{7} \)
So, the best answer for the question, "which one would result in a repeating decimal," includes:
B. \( \frac{1}{9} \)
C. \( \frac{5}{11} \)
D. [tex]\( \frac{3}{7} \)[/tex]
