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Which fractions are equivalent to 0.45‾0.\overline{45}? Select all that apply.\newlineMulti-select Choices:\newline(A) 920\frac{9}{20}\newline(B) 511\frac{5}{11}\newline(C) 4599\frac{45}{99}\newline(D) 45100\frac{45}{100}

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Checkpoint: Rational and irrational numbersright chevron icon

Full solution

Q. Which fractions are equivalent to 0.45‾0.\overline{45}? Select all that apply.\newlineMulti-select Choices:\newline(A) 920\frac{9}{20}\newline(B) 511\frac{5}{11}\newline(C) 4599\frac{45}{99}\newline(D) 45100\frac{45}{100}
  1. Identify decimal as fraction: Identify the repeating decimal 0.45‾0.\overline{45} as a fraction.\newlineTo convert a repeating decimal to a fraction, let x=0.45‾x = 0.\overline{45}. Multiply xx by 100100 (since the repeating part has two digits), so 100x=45.45‾100x = 45.\overline{45}. Subtract xx from 100x100x: 100x−x=45.45‾−0.45‾100x - x = 45.\overline{45} - 0.\overline{45}. This simplifies to 99x=4599x = 45.
  2. Convert decimal to fraction: Solve for xx by dividing both sides by 9999: x=4599x = \frac{45}{99}. Simplify the fraction 4599\frac{45}{99} by dividing the numerator and the denominator by their greatest common divisor, which is 99. So, 4599=511\frac{45}{99} = \frac{5}{11}.

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