How I learned this at GCSE was to multiply by factors of 10.
Example 1What is the fraction equivalent to
0.7˙98˙?
Step 1-
Multiply until you get the same recurring decimal part.
In this case, we need to multiply by 100:
x = 0.798798...
1000x = 798.798798...
See how the decimal part is the same, but the whole number is different?
(
This is why I chose 1000- if I chose 10, the decimal part would be different, e.g. 7.897897... This means we can't subtract easily to get rid of the recurring bit)Step 2- Subtract999x = 798
Step 3- Dividex = 798/999 = 266/333 Final Answer
Example 2What is the decimal equivalent to 0.91919191...
x = 0.919191...
100x = 91.919191...
99x = 91
x = 91/99 Final Answer
Example 3What is the decimal equivalent to 0.123454545...
x = 0.123454545...
100x = 12.34545...
(Do not worry about the extra non-recurring parts, just make sure the recurring parts line up)99x = 12.222
(no recurring)x = 12.222/99 = 12222/99000 = 679/5500 Final Answer
This is more consistent method that works for any recurring decimal.