How do you write \( -20 . \overline{09} \) as a fraction?

To convert \( -20 . \overline{09} \) into a fraction, first separate the integer and the repeating decimal. The integer is -20. Now, focus on the repeating decimal \( .\overline{09} \). Let \( x = 0.\overline{09} \). This means \( x = 0.090909...\). Multiply both sides of the equation by 100 to shift the decimal point two places to the right: \[ 100x = 9.090909... \] Now, subtract the original equation \( x \) from this new equation: \[ 100x - x = 9.090909... - 0.090909... \] This simplifies to: \[ 99x = 9 \] Thus, solving for \( x \) gives: \[ x = \frac{9}{99} = \frac{1}{11} \] Now substituting back, we have \( -20 . \overline{09} = -20 + \frac{1}{11} \). To express this as a single fraction, convert -20 into a fraction with a denominator of 11: \[ -20 = -\frac{220}{11} \] So combined, we get: \[ -\frac{220}{11} + \frac{1}{11} = -\frac{219}{11} \] Therefore, \( -20 . \overline{09} \) as a fraction is \( -\frac{219}{11} \). Remember, repeating decimals are a way of expressing fractions where the digits after the decimal point repeat indefinitely! It’s like a never-ending party of numbers! When you find the fraction, it helps in many areas like math, finance, and science, making calculations easier and precise.