There is a technique for changing recurring decimals to fractions.
Let x = 0.21919191919....... to infinity
10x = 2.19191919191919...to infinity.
1000x = 219.1919191919 to infinity
Do you see what will happen if we subtract 1000x -10x?
1000x = 219.1919191919 to infinity
ul(-10x)" " ul(-2.19191919191919...)to infinity.
990x = 217.0000000000000... to infinity
Solve for x
x = 217/990
Notice the following:
Not all the decimals recur.
The recurring starts after 3 decimal places. This was the reason for multiplying by 1000. The digits must line up so that when you subtract there will be zero's to infinity.
This method can be summarised as follows:
x = (219-2)990 = 217/990
291: " " write down all the digits until they start to recur.
-2: " "subtract the digits that do NOT recur
div 990:" " divide by a 9 for each recurring digit followed by a 0 for each non-recurring digit.
