Step by step
The question
Question: write 0.666 repeating as a fraction.
Look at the digits.
The 6 repeats for ever.
1 Call it x
We do not know the fraction yet.
So we call the number x: x = 0.666 repeating.
Careful!
It is not 6/10.
That is 0.6, which stops.
Ours goes on for ever.
2 Move the point
1 digit repeats.
So we multiply both sides by 10.
That moves the point 1 place to the right.
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 6 for ever.
The tails are exactly the same, so they make zero.
Cross them out.
10x minus x is 9x.
6 minus 0 is 6.
4 Remove the 9
9x means 9 times x.
We want just x.
To remove times 9, we divide by 9.
9 divided by 9 is 1.
So just x is left.
6 and 9 can both be divided by 3.
We can make 6/9 simpler.
It is the same as 2/3.
So x = 2/3.
So 0.666 repeating is 2/3.
5 Check
Let's check: 2 divided by 3.
We get 0.666, and the digits go on the same way.
That is 0.666 repeating.
So the answer is right!
Let's look back.
First: call it x.
Then: move the point.
Then: take away.
Then: remove the 9.
Last: check.