Repeating decimals to fractions
Case: one repeating digit
Example: 0.11111...
Let's define:
x: repeating decimal
x = 0.11111...
Multiplying by 10 at both sides:
10x = 10*0.11111...
10x = 1.1111...
Subtracting x to 10x
10x - x = 1.1111... - 0.11111...
9x = 1
x = 1/9
example: 2.999....
Similarly than before:
x = 2.999....
Multiplying by 10 at both sides:
10x = 29.999...
Subtracting x to 10x:
10x - x = 29.999... - 2.999...
9x = 27
x = 27/9
x = 3
example: 1.2565656...
x = 1.2565656...
Multiplying by 100 at both sides:
100x = 100*1.2565656...
100x = 125.65656...
Subtracting x to 100x:
100x - x = 125.65656... - 1.2565656...
99x = 124.4
x = 124.4/99 = 1244/990 = 622/495