formula for any repeating decimal

Use the following mnemonic:

Examples

1/   `0.dot7 = (7-0)/9 = 7/9`

2/   `0.\dot{1}2\dot{5} = (125 - 0)/999 = 125/999`

3/   `0.\dot{1}\dot{5} = (15-0)/99 = 15/99`

4/   `1.\dot{3}5\dot{2} = 1 + 0.\dot{3}5\dot{2} = 1 + (352 - 0)/999 = (999 + 352)/999 = 1351/999`

5/   `0.1\dot{1}\dot{7} = (117 - 1)/990 = 116/990`

6/   `0.00\dot{6}\dot{9} = (0069 - 00)/9900 = 69/9900`

7/   `0.4\dot{6} = (46 - 4)/90 = 42/90` or `7/15`