Step by step
The question
Question: write 0.2727 repeating as a fraction.
Look at the digits.
The block 2 7 repeats for ever.
1 Call it x
We do not know the fraction yet.
So we call the number x: x = 0.2727 repeating.
Careful!
It is not 27/100.
That is 0.27, 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 2 7 for ever.
The tails are exactly the same, so they make zero.
Cross them out.
100x minus x is 99x.
27 minus 0 is 27.
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.
27 and 99 can both be divided by 9.
We can make 27/99 simpler.
It is the same as 3/11.
So x = 3/11.
So 0.2727 repeating is 3/11.
5 Check
Let's check: 3 divided by 11.
We get 0.2727, and the digits go on the same way.
That is 0.2727 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.