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Which fractions are equivalent to 0.170.1\overline{7}? Select all that apply.\newlineMulti-select Choices:\newline(A) 1690\frac{16}{90}\newline(B) 17100\frac{17}{100}\newline(C) 845\frac{8}{45}\newline(D) 1771,000\frac{177}{1,000}

Full solution

Q. Which fractions are equivalent to 0.170.1\overline{7}? Select all that apply.\newlineMulti-select Choices:\newline(A) 1690\frac{16}{90}\newline(B) 17100\frac{17}{100}\newline(C) 845\frac{8}{45}\newline(D) 1771,000\frac{177}{1,000}
  1. Convert to fraction: Convert 0.1(7)0.1(7) to a fraction. Since 0.1(7)0.1(7) means 0.177777...0.177777..., let x=0.177777...x = 0.177777... Multiplying by 100100, we get 100x=17.7777...100x = 17.7777... Subtracting the original xx from this, 100xx=17.7777...0.177777...100x - x = 17.7777... - 0.177777... results in 99x=17.699x = 17.6. Solving for xx gives 0.1(7)0.1(7)00.
  2. Simplify fraction: Simplify the fraction 17.6/9917.6/99. Since 17.617.6 can be written as 176/10176/10, the fraction becomes 176/990176/990. Simplifying by dividing numerator and denominator by 22, we get 88/49588/495.
  3. Check option (A): Check each option to see if it simplifies to 88495\frac{88}{495}. For (A) 1690\frac{16}{90}, simplifying by dividing numerator and denominator by 22 gives 845\frac{8}{45}.
  4. Check option (B): For (B) 17100\frac{17}{100}, this fraction is already in its simplest form and does not equal 88495\frac{88}{495}.
  5. Check option (C): For (C) 845\frac{8}{45}, this was obtained from simplifying 1690\frac{16}{90}, which we already found to be equivalent to 88495\frac{88}{495}.
  6. Check option (D): For (D) 1771,000\frac{177}{1,000}, simplifying by dividing numerator and denominator by 1,0001,000 would not be correct since 177177 is not divisible by 1,0001,000.

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