Express each repeating decimal as a fraction in lowest terms. $0.\overline{257}$

Verified step by step guidance

1. Let \( x = 0.257257257\ldots \), where the digits 257 repeat indefinitely.
2. Since the repeating block has 3 digits, multiply \( x \) by \( 10^3 = 1000 \) to shift the decimal point three places to the right: \( 1000x = 257.257257257\ldots \).
3. Set up the equation by subtracting the original \( x \) from this new expression to eliminate the repeating decimal part: \( 1000x - x = 257.257257257\ldots - 0.257257257\ldots \).
4. Simplify the subtraction to get \( 999x = 257 \), since the repeating decimals cancel out.
5. Solve for \( x \) by dividing both sides by 999: \( x = \frac{257}{999} \). Then, simplify the fraction to its lowest terms by finding the greatest common divisor (GCD) of 257 and 999 and dividing numerator and denominator by it.

Key Concepts

Repeating Decimals

A repeating decimal is a decimal number in which a sequence of digits repeats infinitely. For example, 0.257̅ means the digits '257' repeat endlessly. Understanding this pattern is essential to convert the decimal into a fraction.

Algebraic Representation of Repeating Decimals

To convert a repeating decimal to a fraction, represent the decimal as a variable (e.g., x), then multiply by a power of 10 to shift the decimal point to align repeating parts. Subtracting these equations eliminates the repeating portion, allowing you to solve for x as a fraction.

Simplifying Fractions to Lowest Terms

After finding the fraction equivalent of the repeating decimal, simplify it by dividing numerator and denominator by their greatest common divisor (GCD). This ensures the fraction is expressed in its simplest form, which is a standard requirement in algebra.