More repetition fun in division

Mental Calculation, Mathematics, and Puzzles
1

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

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Division Mentally
Fastest way to learn mental math
2

Alternative approach for 13ths:

First, for irregular fractions, convert them into proper fractions. Use your favorite method here. The idea is that you want the numerator above 13 to range from 1 to 12. For example, 100/13 = 7 and 9/13, and 231/13 = 17 and 10/13.

To work from this point, you’re going to multiply the numerator by 77. It’s best to do this in 2 stages: Multiply by 7, then multiply by 11 (this can seem hard, but multiplying by 11 mentally isn’t difficult if you know the classic trick).

So, in 9/13 the numerator is 9. First, we multiply that by 7 to get 63. Then we multiply that result by 11 to get 693.

The next step is to simply subtract 1 from the result you get. 693 - 1 = 692.

Finally, to get the next 3 numbers, just subtract that result from 999 (think of it as subtracting each digit from 9): So, 999 - 692 = 307.

That means the digits after the decimal point are .692307 repeating, as in .692307692307692307… and so on.

More detail on this approach in 3 parts here:
THIRTEENTHS (PART 1/3): SECRETS OF THE MATHEMATICAL NINJA
THIRTEENTHS (PART 2/3): SECRETS OF THE MATHEMATICAL NINJA
THIRTEENTHS (PART 3/3): SECRETS OF THE MATHEMATICAL NINJA

Oh, and once you’ve mastered 13ths, then 26ths aren’t much of an obstacle:
THE MATHEMATICAL NINJA AND THE TWENTY-SIXTHS

That last approach works for even denominators which aren’t divisible by 4, as long as you’re comfortable dividing by half that denominator. So, if you’re comfortable dividing by 7, you can adapt the trick to 14ths. Comfortable dividing by 9? Then 18ths won’t be a problem. Dividing by 11 means you can divide by 22, as well!

3

Excellent!
I love Colins’ work btw. Great blog.

4

@Kinma
For learning repetation of 7
I made a little logic to learn this by counting number system.

I LOVE YOU * 999
1 4 3 * 999 = 142857

And you know how to multiply by 999 it’s just simple in mind.

5

I love it!

Personally I use double 7 → repeat (double answer)

So double 7 = 14 (first 2 digits).
Double answer = 28 (next 2 digits).
Double answer = 56 (next 2 digits).
Keep in mind that in the next step, double 56 is 112, so 56 becomes 57 because of the carry.

You can continue (but now you have a carry from the next step) or remember that it repeats after 6 digits.

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6

Every time I explain it as doubles, there is always someone who says “but double of 28 is 56” before I get a chance to mention the 128 carry to the left. :wink:

\begin{array}{rrrrr} &{\color{green}7} &1 &4 \\ &2 &8 &{\color{green}5} \\ \hline &9 &9 &9 \end{array}

…they also add to 9, so by putting the 28 onto the second row, you can avoid the 57-discussion, because you don’t need the unit digit.