Technique explanation
Recurring decimals can be converted back to fractions using a subtraction-normalization method. For pure recurring decimals, the denominator consists of as many 9s as there are digits in the period. For mixed recurring decimals, the numerator is the (whole part) minus (non-recurring part), and the denominator uses 9s for recurring digits and 0s for non-recurring decimal digits. The algebraic method remains the standard fallback.
Standard fallback: The algebraic method: let x = 0.3636..., then 100x = 36.3636..., so 99x = 36, giving x = 36/99.
Worked examples
Example 1: Convert 0.36 recurring to a fraction.
1. The recurring part is 36 (2 digits).
2. The denominator is 99 (two 9s).
3. Fraction: 36/99. Simplify by 9: 4/11.
Check: 4 ÷ 11 = 0.3636...
Check: Perform division: 4 ÷ 11 = 0.3636...
Result: 4/11.
Example 2: Convert 0.16 recurring (only 6 repeats) to a fraction.
1. Non-recurring: 1. Recurring: 6.
2. Numerator: 16 - 1 = 15.
3. Denominator: One 9 (for 6) and one 0 (for 1) = 90.
4. Fraction: 15/90. Simplify by 15: 1/6.
Check: 1 ÷ 6 = 0.1666...
Check: Perform division: 1 ÷ 6 = 0.1666...
Result: 1/6.
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