Step by step
The question
Question: What is the 60th digit after the decimal point of 3/7?
1 Find the block
30 divided by 7 is 4, and 2 is left.
We divide 3 by 7 the long way.
Each time we bring down a 0.
20 divided by 7 is 2, and 6 is left.
60 divided by 7 is 8, and 4 is left.
40 divided by 7 is 5, and 5 is left.
50 divided by 7 is 7, and 1 is left.
10 divided by 7 is 1, and 3 is left.
3 left: that is where we started.
So the digits repeat from here.
So 3/7 is 0 point 4 2 8 5 7 1, and the same 6 digits again and again.
The block is 6 long.
2 Divide
The block comes again and again.
So we divide by its length.
What is left over tells us the place in the block.
6 times 10 is 60.
So 10 full blocks take us to the 60th digit.
60 minus 60 is 0 more.
3 Count into the block
Nothing is left over.
So the 60th is the end of a block: the last one, 1.
4 Check
Let's check by counting on.
A block ends at the 54th digit.
Then we count on: 4, 2, 8, 5, 7, 1.
The 60th is 1.
So it is right!
Let's look back.
First: find the block.
Then: divide.
Then: count into the block.
Last: check.