How do you turn 0.11111... (recurring) into a fraction

Let's look at what makes this question more difficult than, say, 0.5 or 0.01: as the decimal is recurring, you can't just multiply and divide by a big number to get a fraction. Example: multiply 0.01 times 100, then divide by 100. You get 1/100 which is your fraction. However, if you multiply 0.11111... by 100 you get 11.11111... (still recurring). This just means we require an additional step to solve this. Let's try to get rid of the recurring decimal with some simple operations: let x = 0.111111... then 10x = 1.111111... 10x - x = 1.111111... - 0.111111... = 1 and: 10x - x = 9x So, we can write 9x = 1. Hence, x = 1/9 is your fraction. We can expand this method for all sorts of recurring decimals, like 0.12121212... or 0.426426426... with slight changes in our method, which I would like you to try and find, (Hint: remember we want to get rid of the recurring decimal).