To determine which of the given fractions result in a repeating decimal when converted to their decimal forms, let's analyze each option:
### Fraction A: [tex]\( \frac{3}{4} \)[/tex]
- Step 1: Convert [tex]\( \frac{3}{4} \)[/tex] to a decimal.
- Calculation: [tex]\( \frac{3}{4} = 0.75 \)[/tex]
- Conclusion: [tex]\( 0.75 \)[/tex] is a terminating decimal and hence not repeating.
### Fraction B: [tex]\( \frac{5}{11} \)[/tex]
- Step 1: Convert [tex]\( \frac{5}{11} \)[/tex] to a decimal.
- Calculation: [tex]\( \frac{5}{11} \approx 0.454545\ldots \)[/tex]
- Conclusion: The decimal [tex]\( 0.454545\ldots \)[/tex] repeats (specifically, "45" repeats).
### Fraction C: [tex]\( \frac{1}{9} \)[/tex]
- Step 1: Convert [tex]\( \frac{1}{9} \)[/tex] to a decimal.
- Calculation: [tex]\( \frac{1}{9} \approx 0.111111\ldots \)[/tex]
- Conclusion: The decimal [tex]\( 0.111111\ldots \)[/tex] repeats (specifically, "1" repeats).
### Fraction D: [tex]\( \frac{3}{7} \)[/tex]
- Step 1: Convert [tex]\( \frac{3}{7} \)[/tex] to a decimal.
- Calculation: [tex]\( \frac{3}{7} \approx 0.428571428571\ldots \)[/tex]
- Conclusion: The decimal [tex]\( 0.428571428571\ldots \)[/tex] repeats (specifically, "428571" repeats).
### Summary:
- [tex]\( \frac{3}{4} \)[/tex] = [tex]\( 0.75 \)[/tex] (terminating)
- [tex]\( \frac{5}{11} \)[/tex] ≈ [tex]\( 0.454545... \)[/tex] (repeating)
- [tex]\( \frac{1}{9} \)[/tex] ≈ [tex]\( 0.111111... \)[/tex] (repeating)
- [tex]\( \frac{3}{7} \)[/tex] ≈ [tex]\( 0.428571428571... \)[/tex] (repeating)
Therefore, the fractions [tex]\( \frac{5}{11} \)[/tex], [tex]\( \frac{1}{9} \)[/tex], and [tex]\( \frac{3}{7} \)[/tex] result in repeating decimals.
### Answer:
B. [tex]\( \frac{5}{11} \)[/tex], C. [tex]\( \frac{1}{9} \)[/tex], and D. [tex]\( \frac{3}{7} \)[/tex].
