How do you turn 0.1 repeating into a fraction?

Answer 1

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 = 1/9

Therefore 0.111... = 1/9

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To convert any repeating decimal to a fraction:

1. 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;
2. Multiply the repeating decimal by the power of 10 found in step {1};
3. Subtract the original number from number calculated in step {2}
4. 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;
5. 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 normal

For 0.181818... this gives:

1. 2 digits repeat, so power of 10 is 2 → 10² = 100
2. 0.181818... × 100 = 18.181818...
3. 18.181818... - 0.181818... = 18
4. 18/(100-1) = 18/99
5. 18/99 = (2×9)/(11×9) = 2/11

Thus 0.181818... = 2/11

For 0.1666... this gives:

1. 1 digit repeats, so power of 10 is 1 → 10¹ = 10
2. 0.1666... × 10 = 1.666...
3. 1.666... - 0.1666... = 1.5
4. 1.5/(10-1) = 1.5/9
5. 1.5/9 = (1.5×10)/(9×10) = 15/90 = (15×1)/(15×6) = 1/6

Thus 0.1666... = 1/6