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Which fractions are equivalent to 0.630.\overline{63}? Select all that apply.\newlineMulti-select Choices:\newline(A) 6399\frac{63}{99}\newline(B) 711\frac{7}{11}\newline(C) 611\frac{6}{11}\newline(D) 63100\frac{63}{100}

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Q. Which fractions are equivalent to 0.630.\overline{63}? Select all that apply.\newlineMulti-select Choices:\newline(A) 6399\frac{63}{99}\newline(B) 711\frac{7}{11}\newline(C) 611\frac{6}{11}\newline(D) 63100\frac{63}{100}
  1. Convert to Fraction: Step 11: Convert 0.630.\overline{63} to a fraction.\newline0.630.\overline{63} means 0.636363...0.636363..., which is a repeating decimal.\newlineTo convert it to a fraction, let x=0.636363...x = 0.636363...\newlineThen, 100x=63.636363...100x = 63.636363...\newlineSubtracting these, 99x=6399x = 63\newlineSo, x=6399.x = \frac{63}{99}.
  2. Simplify Fraction: Step 22: Simplify 6399\frac{63}{99}. The greatest common divisor (GCD) of 6363 and 9999 is 99. Divide both numerator and denominator by 99: 63÷9=763 \div 9 = 7, 99÷9=1199 \div 9 = 11. Thus, 6399\frac{63}{99} simplifies to 711\frac{7}{11}.
  3. Check Options: Step 33: Check each option.\newline(A) 6399\frac{63}{99} is equivalent to 711\frac{7}{11}, which we found is equivalent to 0.{63}0.\{63\}.\newline(B) 711\frac{7}{11} is the simplified form of 6399\frac{63}{99}, so it is equivalent.\newline(C) 611\frac{6}{11} needs to be checked.\newlineTo check, convert 611\frac{6}{11} to a decimal: 6÷110.545454...6 \div 11 \approx 0.545454..., which is not 0.636363...0.636363...\newline(D) 63100\frac{63}{100} as a decimal is 711\frac{7}{11}00, which is not repeating like 0.636363...0.636363...

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