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Write the repeating decimal as a fraction.
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Q. Write the repeating decimal as a fraction.
- Identify Repeating Pattern: Identify the repeating pattern in the decimal. The repeating pattern is , which repeats indefinitely after the decimal point.
- Express as Infinite Sum: Express the repeating decimal as an infinite sum of its repeating parts. Since the pattern starts after the second decimal place, we can write it as:
- Convert to Fractions: Convert each term in the sum into a fraction. The first term is , the second term is , and so on. This gives us:
- Recognize Geometric Series: Recognize that the sum forms a geometric series with the first term and the common ratio .
- Use Sum Formula: Use the formula for the sum of an infinite geometric series, which is , to write the repeating decimal as a fraction. Substitute and into the formula.Sum =
- Simplify Expression: Simplify the expression by finding a common denominator and performing the division.Sum = Sum =
- Multiply Numerators and Denominators: Multiply the numerators and denominators to get the fraction.Sum = Sum =
- Reduce to Simplest Form: Reduce the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor, which is .
