In Exercises 45–50, express each repeating decimal as a fraction in lowest terms. 0.47 ̅ (repeating 47) = 47/100 + 47/10,000 + 47/1,000,000 + ...

Verified step by step guidance

1. Let x = 0.474747... (where 47 is the repeating part).
2. Multiply both sides of the equation by 100 to shift the decimal point two places to the right: 100x = 47.474747...
3. Subtract the original equation (x = 0.474747...) from this new equation (100x = 47.474747...) to eliminate the repeating part: 100x - x = 47.474747... - 0.474747...
4. Simplify the equation: 99x = 47.
5. Solve for x by dividing both sides by 99: x = 47/99.

Key Concepts

Repeating Decimals

Repeating decimals are decimal numbers in which a digit or a group of digits repeats infinitely. For example, 0.47̅ means that '47' continues indefinitely. Understanding how to represent these decimals as fractions is crucial, as it involves recognizing the pattern in the digits and applying mathematical techniques to convert them into a fraction.

Geometric Series

A geometric series is a series of terms where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the case of repeating decimals, the conversion to a fraction often involves identifying the series formed by the repeating part and using the formula for the sum of an infinite geometric series to find its fractional representation.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its lowest terms by dividing the numerator and denominator by their greatest common divisor (GCD). This step is essential after converting a repeating decimal to a fraction, as it ensures that the final answer is expressed in the simplest form, making it easier to understand and use in further calculations.