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

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Q. Which fractions are equivalent to 0.630.\overline{63}? Select all that apply.\newlineMulti-select Choices:\newline(A) 711\frac{7}{11}\newline(B) 611\frac{6}{11}\newline(C) 63100\frac{63}{100}\newline(D) 6399\frac{63}{99}
  1. Convert to fraction: Step 11: Convert 0.630.\overline{63} to a fraction.\newline0.630.\overline{63} means 0.636363...0.636363..., a repeating decimal. To convert it to a fraction, let x=0.636363...x = 0.636363... Multiplying by 100100 (since the repeating part has two digits), we get 100x=63.636363...100x = 63.636363... Subtracting the original xx from this, 100xx=63.636363...0.636363...100x - x = 63.636363... - 0.636363... results in 99x=6399x = 63. 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. Dividing both numerator and denominator by 99, we get 6399=711.\frac{63}{99} = \frac{7}{11}.
  3. Check options: Step 33: Check each option.\newline(A) 711\frac{7}{11} is equivalent to 6399\frac{63}{99}, as calculated.\newline(B) 611\frac{6}{11} is not equivalent because 611=54.545454...\frac{6}{11} = 54.545454... which is different from 0.636363...0.636363...\newline(C) 63100\frac{63}{100} is not equivalent because 63100=0.63\frac{63}{100} = 0.63, which is a terminating decimal and not repeating like 0.636363...0.636363...\newline(D) 6399\frac{63}{99} is equivalent, as shown in Step 11.

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