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Write the repeating decimal as a fraction.
Full solution
Q. Write the repeating decimal as a fraction.
- Rephrase Problem: Let's first rephrase the problem into a single ＂How can we express the repeating decimal as a fraction?＂
- Identify Repeating Pattern: Identify the repeating pattern in the decimal. The digits ＂＂ repeat indefinitely, so we can write the decimal as
- Express Terms as Fractions: Express each term in the pattern as a fraction. The first term is , which is . The second term is , which is , and so on. This gives us the series:
- Recognize Geometric Series: Recognize that the series forms a geometric series, where each term is times the previous term. The first term is , and the common ratio is .
- Use Sum Formula: Use the formula for the sum of an infinite geometric series, which is , where is the first term and is the common ratio. Substitute and into the formula.
- Calculate Sum: Calculate the sum of the series: .
- Simplify Fraction: Simplify the fraction . Both the numerator and the denominator are divisible by . Dividing both by gives us .
