In Exercises 45–50, express each repeating decimal as a fraction in lowest terms. 0.257 ̅ (repeating 257)

Verified step by step guidance

1. Identify the repeating part of the decimal, which is 257.
2. Let x = 0.257\overline{257}, which means x = 0.257257257... .
3. Multiply x by 1000 to shift the decimal point three places to the right: 1000x = 257.257257... .
4. Subtract the original x from this new equation: 1000x - x = 257.257257... - 0.257257... .
5. Solve the resulting equation: 999x = 257, then express x as a fraction: x = \frac{257}{999}.

Key Concepts

Repeating Decimals

Repeating decimals are decimal numbers in which a digit or a group of digits repeats infinitely. For example, in the decimal 0.257̅, the digits '257' repeat indefinitely. Understanding how to identify and represent these decimals is crucial for converting them into fractions.

Conversion of Decimals to Fractions

To convert a repeating decimal to a fraction, one typically sets the decimal equal to a variable, then manipulates the equation to eliminate the repeating part. This often involves multiplying the equation by a power of 10 that corresponds to the length of the repeating segment, allowing for subtraction and simplification to find the fraction.

Lowest Terms

A fraction is in lowest terms when the numerator and denominator have no common factors other than 1. After converting a repeating decimal to a fraction, it is essential to simplify it by dividing both the numerator and denominator by their greatest common divisor (GCD) to ensure the fraction is expressed in its simplest form.