Step by step
The question
Question: write 1.0909 repeating as a fraction.
Look at the digits.
The block 0 9 repeats for ever.
1 Call it x
We do not know the fraction yet.
So we call the number x: x = 1.0909 repeating.
Careful!
It is not 109/100.
That is 1.09, which stops.
Ours goes on for ever.
2 Move the point
2 digits repeat.
So we multiply both sides by 100.
That moves the point 2 places to the right, one whole block.
3 Take away
Now we take the first one away from the new one.
Both sides, so they stay equal.
After the point, both go on with 0 9 for ever.
The tails are exactly the same, so they make zero.
Cross them out.
100x minus x is 99x.
109 minus 1 is 108.
4 Remove the 99
99x means 99 times x.
We want just x.
To remove times 99, we divide by 99.
99 divided by 99 is 1.
So just x is left.
108 and 99 can both be divided by 9.
We can make 108/99 simpler.
It is the same as 12/11.
So x = 12/11.
So 1.0909 repeating is 12/11.
It is bigger than 1, so the top is bigger than the bottom.
5 Check
Let's check: 12 divided by 11.
We get 1.0909, and the digits go on the same way.
That is 1.0909 repeating.
So the answer is right!
Let's look back.
First: call it x.
Then: move the point.
Then: take away.
Then: remove the 99.
Last: check.