Which simplified fraction represents \( 0 . \overline{45} \) ? \( \frac{45}{99} \) \( \frac{99}{45} \) \( \frac{511}{5} \)

To convert the repeating decimal \( 0.\overline{45} \) into a fraction, we can set \( x = 0.\overline{45} \). Multiplying both sides by 100 to move the decimal point two places to the right gives us \( 100x = 45.\overline{45} \). Now, subtract the original \( x = 0.\overline{45} \) from this equation: \[ 100x - x = 45.\overline{45} - 0.\overline{45} \] This simplifies to: \[ 99x = 45 \] Dividing both sides by 99 yields: \[ x = \frac{45}{99} \] When simplified, \( \frac{45}{99} \) reduces to \( \frac{5}{11} \). In historical contexts, fractions were a fundamental aspect of early mathematics, long before calculators made calculations easier. Ancient Egyptians used fractions for measurements in building construction, while Babylonian scribes employed clay tablets to record various fraction values. In the real world, understanding how to convert repeating decimals to fractions can help with various tasks like calculating interest rates, analyzing statistical data, and simplifying calculations in finance. This skill proves invaluable in making quick and accurate assessments in everyday situations, like budgeting or when shopping for deals!