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Which fractions are equivalent to 0.17‾0.1\overline{7}? Select all that apply.\newlineMulti-select Choices:\newline(A) 1771,000\frac{177}{1,000}\newline(B) 1690\frac{16}{90}\newline(C) 845\frac{8}{45}\newline(D) 17100\frac{17}{100}

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Checkpoint: Rational and irrational numbersright chevron icon

Full solution

Q. Which fractions are equivalent to 0.17‾0.1\overline{7}? Select all that apply.\newlineMulti-select Choices:\newline(A) 1771,000\frac{177}{1,000}\newline(B) 1690\frac{16}{90}\newline(C) 845\frac{8}{45}\newline(D) 17100\frac{17}{100}
  1. Convert to fraction: Convert 0.1(7)0.1(7) to a fraction by recognizing it as 0.170.17 repeating. Let x=0.171717…x = 0.171717\ldots Multiply xx by 100100 to shift the decimal point: 100x=17.171717…100x = 17.171717\ldots Subtract the original xx from this equation: 100x−x=17.171717…−0.171717…100x - x = 17.171717\ldots - 0.171717\ldots This simplifies to 99x=1799x = 17.
  2. Solve for x: Solve for x by dividing both sides by 9999: x=1799x = \frac{17}{99}. This fraction represents the decimal 0.1(7)0.1(7).
  3. Check options: Check each option to see if it simplifies to 1799\frac{17}{99}:\newline(A) 1771,000\frac{177}{1,000} does not simplify to 1799\frac{17}{99}.\newline(B) 1690\frac{16}{90} simplifies to 845\frac{8}{45}, which needs further checking.\newline(C) 845\frac{8}{45} simplifies to 845\frac{8}{45}, which is the same as 1690\frac{16}{90}.\newline(D) 17100\frac{17}{100} does not simplify to 1799\frac{17}{99}.
  4. Simplify 845\frac{8}{45}: Simplify 845\frac{8}{45} to check if it equals 1799\frac{17}{99}. Divide both numerator and denominator by their GCD, which is 11, so 845\frac{8}{45} remains 845\frac{8}{45}. This does not equal 1799\frac{17}{99}.

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