A recurring decimal can be converted to a fraction using powers of 10.
For example: `0. dot3 dot2` could be written as 0.3232323232....
Set x as 0.3232323232...
The number recurs every second digit.
Multiply both x and the number by 102 such that 100x = 32.32323232...
Subtract the original equation from the second equation:
100x = 32.32323232...
x = 0.3232323232.... , and subtract:
99x = 32, and make a fraction such that
`x = frac(32)(99)`.
If the number repeats every digit, multiply by 10; every third digit, multiply by 103 etc
Convert `0.dot1 2 dot3` to a fraction.
| `x` | `= 0.123123123...` | |
| Recurs every 3 digits | `` | `` |
| Multiply by 103 | `1000x` | `=123.123123123...` |
| Subtract equation | `999x` | `=123` |
| `x` | `=frac(123)(999)` | |
| Simplify: | `x` | `=frac(41)(333)` |
Answer: `frac(41)(333)`
Convert `2.dot4 dot5` to a fraction.
| `x` | `= 2.45454545...` | |
| Recurs every 2 digits | ||
| Multiply by 102 | `100x` | `= 245.454545...` |
| Subtract | `99x` | `= 243`. |
| `x` | `= frac(243)(99)`. | |
| Simplify | `x` | `= 2frac(5)(11)`. |
Answer: 2`frac(5)(11)`