Let’s do 100/13 and forget about the digital separator for now. We focus on the digits since we know where the separator must come, right?

If you are new to doing division with negative remainders, see here: Making two digit (or more) mental division easier by using negative remainders

100 goes about 7 times into 13. Let’s do the 2 step process again:

100 - 70 = 30. 30 - 21 = 9

Answer so far: 7 R 9

9 -> 90. 90 goes about 7 times into 13:

90 - 70 = 20. 20 - 21 = -1

Answer so far: 77 R -1

-1 -> -10. 10 goes zero times into 13.

Answer so far: 770 R -10

-10 -> -100.

-100? Wait. This is interesting!

-100 is - except for the minus sign - the amount we just started with.

So, instead of calculating digit by digit, let’s take the work we just did for 100/13, apply this for -100 and work with ‘770’ as a group of digits.

If 100 / 13 = 770 R -10, then -100/13 = -770 R 10

So we need to mentally subtract ‘770’ from 770.

Try to see this as 770 - 0.770.

One easy way of mentally doing the subtraction is to subtract 1 from 770 = 769 and then to add the complement: 0.23 = 769.230.

We know now that 100/13 = 769230 R 10

In the next step the remainder will be 100 so this whole process now repeats:

769230769230769230769230769230769230769230769230…

By just calculating the first 3 digits and a complement we have calculated 100/13 as: 7.69230769230769230769230769230769230769230769230…

If this happens the first part of the repeating digits added to the second part add up to a repeating set of 9’s:

769

230

___+

999

In my last examples I always took 100 as the dividend.

This might give the impression that the repetition and the 2 halves that add up to a set of nines only work with 100 as dividend.

This is not true.

An example: 231/13 = 17.769230769230769…

Sometimes the repetition starts at a different point in the sequence:

237/13 = 18.230769230769230769230769230

A little more work is a division by 29.

1/29 = 0.34482758620689655172413793103448275862068965517241379310

It might not be immediately obvious, but we see the same effect here:

34482758620689 65517241379310 34482758620689 65517241379310

34482758620689

65517241379310

______________+

99999999999999

1/47 = 2127659574468085106382978723404255319148936170

21276595744680851063829

78723404255319148936170

______________________+

999999999999999999999