How do you turn 0.1 repeating into a fraction? - Answers

Let the fraction be q; then:{1} q = 0.111...Multiply both sides by 10:{2} 10q = 1.111...Subtract {1} from {2} to get:10q - q = 1.111... - 0.111...→ 9q = 1→ q = 1/9Therefore 0.111... = 1/9--------------------------------------To convert any repeating decimal to a fraction:Count the number of repeating digits and use it as the power of 10, eg for 0.181818... there are 2 repeating digits (18), so the power is 2 giving 10² = 100; for 0.1666... there is 1 repeating digit (the 6), os the power is 1 giving 10¹ = 10;Multiply the repeating decimal by the power of 10 found in step {1};Subtract the original number from number calculated in step {2}Put the result of step {3} as the numerator over 1 less than the power of 10 from step {1} as the denominator; the denominator consists of the digit 9 repeated the number of times of the power of 10;Simplify the fraction.The fraction created may not be a "proper" fraction with a decimal in the numerator; this is not a problem: multiply top and bottom by the required power of 10 to remove the decimal point and then simplify as normalFor 0.181818... this gives:2 digits repeat, so power of 10 is 2 → 10² = 1000.181818... × 100 = 18.181818...18.181818... - 0.181818... = 1818/(100-1) = 18/9918/99 = (2×9)/(11×9) = 2/11Thus 0.181818... = 2/11For 0.1666... this gives:1 digit repeats, so power of 10 is 1 → 10¹ = 100.1666... × 10 = 1.666...1.666... - 0.1666... = 1.51.5/(10-1) = 1.5/91.5/9 = (1.5×10)/(9×10) = 15/90 = (15×1)/(15×6) = 1/6Thus 0.1666... = 1/6