Step by step
The question
Question: write 0.1666 repeating as a fraction.
Look at the digits.
1 comes once.
Then 6 repeats for ever.
1 Call it x
We do not know the fraction yet.
So we call the number x: x = 0.1666 repeating.
Careful!
It is not 16/100.
That is 0.16, which stops.
Ours goes on for ever.
2 Move the point
First we move the point past the 1, which does not repeat.
Times 10 moves the point 1 place to the right.
We do it to both sides, so they stay equal.
Then one block more: times 100 moves it 2 places.
Now both numbers have the same tail after the point.
3 Take away
Now we take the smaller one away from the bigger 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.
100x minus 10x is 90x.
16 minus 1 is 15.
4 Remove the 90
90x means 90 times x.
We want just x.
To remove times 90, we divide by 90.
90 divided by 90 is 1.
So just x is left.
15 and 90 can both be divided by 15.
We can make 15/90 simpler.
It is the same as 1/6.
So x = 1/6.
So 0.1666 repeating is 1/6.
5 Check
Let's check: 1 divided by 6.
We get 0.1666, and the digits go on the same way.
That is 0.1666 repeating.
So the answer is right!
Let's look back.
First: call it x.
Then: move the point.
Then: take away.
Then: remove the 90.
Last: check.