Mental Calculation - Division - Resources?

Has anyone built up a nice repository of how to think about division online or in the books? At this point I have mostly just attempted to improve my multiplication skills in order to better handle this but would appreciate any good reference material to study or work through.

Thanks,
R.

This is a difficult part of mental calculation to find books for. Some books have a small portion devoted to division.
It is difficult to find alternative algorithms. So we have to make them ourselves.

What kind of division are you looking for? Division by 2 digit numbers for example? General algorithms?

What helps for me is to build a table of the divisor in my mind.
Let’s say we calculate 1/21. I do a quick run of the table of 21, so:
21
42
63
84
105
126
147
168
189
210

This helps later in the process.

Then I go: 1 becomes 100, 84 fits into this. Here is why I do a run of the table. Since I just ran the table of 21 I have the numbers 84 and 105 in the front of my mind.
Calculate the remainder: 100-84=16. 16 becomes 160. 147 fits into this. Answer so far: 0.047.
Remainder: 160-147=13. 13 becomes 130. 126 fits. Answer 0.0476.
Since 4(0) is close to 42 we know the rest of the answer is almost 2. So we can decide to stop here with an answer of (slightly less than) 0.04762. Or we continue.

These are all small steps that are easy to do.
There is another technique to work out 1/21. Let me know if you are interested.

Thanks… I’m around grade 6/7 with mental division so that generally means 3-4 digit divided by 2-3 digits with randomly placed decimal points. Like multiplication some is trivially easy and others result in stumbling.

In practice I have been visually estimating then multiplying to build my result. When my knowledge of the multiplicands is crisp this goes quickly. Conversely when it is not it doesn’t. Occasionally I see opportunities to factor and simplify ratios. If I don’t have the ratios well understood I am confused by the time I continue with the mental calculation.

I’ll give you another way of dividing. One that is less well known and is interesting for dividing by multiple digits.

An example will make it clear:
Let’s say we need to know 100 / 21.
What we do is we divide by 20, easy peasy, and subtract a correction in the next step.

100 / 20 is 5. However, this leaves us with no remainder and in the next step we are going to subtract something, which takes us into negative territory. This is to be avoided, so we will use 4 instead of 5:

100 / 20 = 4 R 20
Now the correction: 4 R 20 - 4 X 1 = 4 R 16.
This is correct: 4 X 21 = 84 and 84 + 16 = 100.

We continue. the remainder of 16 becomes 160 and this fits 8 times into 20.
However; since we avoid negative remainders, we will use 7: 160 - 140 = 20.
So we have 47 R 20. Now the correction: 20 - 7x1 = 13.
Our answer so far: 47 R 13.

13 becomes 130 and this fits 6 times into 20: 130 - 120 = 10.
Answer: 476 R 10. correction: 6 X 1 = 6 and 10 - 6 = 4.
Answer: 476 R 4.
4 becomes 40 and this fits 2 times into 20.
Answer 4762 R 0. Correction: -2X1 = -2 so our answer is 4762 R -2.
Since our remainder is very small we can stop or use 4761 R 19 as the answer and continue from there.

Compare this to the previous post where we calculated 1/21.

Now, let’s do the 4 digits divided by 3 digits that Robert needs to do, using the previous method. Let’s take something difficult:

4567 / 686

Let’s start with:
456 / 68

68 is almost 70 and 6 X 70 = 420, so our first number is 6.
If 6 X 70 = 420 then 6 X 68 = 420 - 12 = 408.

456 - 408 = 48.

Add the ‘7’ from 4567 to our remainder: 487.
Now do the correction 6 X 6 = 36 and 487 - 36 = 451

In other words 4567 / 686 = 6 R 451.

We continue with the remainder of 451.
We can use the 6 X 68 = 408 again from the previous steps:

451 - 408 = 43.

The correction. We add a zero, since we used all digits of 4567. This also means the decimal point goes here. The remainder is now 430
430 - 36 = 394.

394 is slightly less than 408, so we use 5 instead of the 6 from the previous 2 steps:

5 X 68 = 340. 394 - 340 = 54.
Add a zero: 540.
Do the correction: 540 - 5 X 6 = 510.

If we take 70 instead of 68 we can guess that we can do this almost 7 times: 7 X 70 = 490. Since 70 is more than 68 we can first subtract the 490 from 510: 510 - 490 = 20. We have now subtracted too much, since we need to subtract 7 X 68 instead of 7 X 70. So we add 2 X 7 = 14.
20 + 14 = 34.

We add a zero: 340
Now we do the correction from 680 to 686:
340 - 7 X 6 = 340 - 42 = 298.

From 4 X 70 = 280, we can guess that our next number is a 4, so we can stop here.

Our answer is: 6.6574…

All calculations are simple. We never have to work out things like 7 X 686. We essentially work out 7 X 70, then subtract 7 X 2, to get to 7 X 68, we add a zero, so we are at 7 X 680 and in the next step do the correction to go to 7 X 686.

In other words, we never do more than 1X1 calculations.
If you practice this a couple of times you can easily do these kinds of calculations mentally.

Thanks, as usual, Kinma. :slight_smile:

I’m going to sit down and work through this a bit this afternoon.
It’s funny learning arithmetic at 50. Thinking about math and doing it are remarkably different things. The mechanics are so darn simple. With paper/pen, calculator, python/matlab we can do calculations that we have little hope of grasping intellectually. Load up the library, run a function, press a button and the answer pops out. An accumulation of ideas, each relying on previous individuals works. Is a mathematician a historian? a philosopher? a calculator? a scientist?.. It’s all very grey.

Starting to have too much fun again… Time for another coffee and more practice. Scope creep is evil.

Hi Kinma,

Is this an example of a Vedic Maths, it looks like similar to a technique that I’m trying to learn.

Hi Tiger,

Can you describe that technique?

Would love to hear and compare differences.

The technique looks similar to this one a few minutes into the video https://www.youtube.com/watch?v=fhHc4qk9i-Q

Looks like it, yes. In the video a system is presented without the reasons why it is the way it is.
But read my posts and you will understand.

Your posts are very helpful in understanding this method of division which is completely new to me. I have found it challenging as it taxes memory because you need to keep track of answers, corrections and complications like negative numbers. I have put in a few hours and I am starting to get more of these correct but it usually has to be done on paper whereas I prefer to do it all mentally. For me I don’t need complete accuracy when dividing large numbers but they do need to be close. I am prepared to sacrifice a little accuracy for the ability to do it all mentally.

Thank you!

Let’s do a simple one:
100 / 29
We will start with 100/30. 100 goes 3 times into 30. Remainder is 10.
Now change the remainder because we are dividing by 29 instead of 30.

Visualise this by seeing 100 apples and divide them over 30 baskets.
You put 3 apples into each basket and keep the last 10 as remainder.
Now, since we are actually dividing by 29 instead of 30, you take the 3 apples from the last basket and put them with the rest of the remainder.

So first step is 100/29 = 3R13.
Next step 130 / 30 = 4R10, so 130/29 = 4R14
3rd step: 140/30 = 4R20, so 140/29 = 4R24
4th step: 240/30 = 8R0, so 240/29 = 8R8.
5th step: 80/30 = 2R20, so 240/29 = 2R22.
etc.

Answer so far: 3,4482
see how quick this is?

Thank you Kinma for that detailed and visual explanation, that really helps to see what is going on. I really like the method and it works beautifully for certain questions where there aren’t complications that involve carrying particularly where there are negative numbers involved. The questions that give me problems are these types: 235813 divided by 64 and 35609 divided by 37.

Here we go . Use 60 as the round number to divide with.

60 goes almost 4 times into 235. So we take 3 times. 60X3 = 180.
First subtract 180 from 235 = 55.
Then another 3x4=12 from 55 to get 43.

Add the ‘8’: 438
60 goes 7 times into 438. 7X60=420.
438-420 = 18.
7X4=28. 18-28 = minus 10.
This presents us with an issue: negative remainders. So back up a step and take 6 times instead of 7.

6X60=360.
438-360 = 78.
6X4=24. 78-24 = 54.

Add the ‘1’: 541.
9 times 60 = 540 leaves a remainder of one, so after the correction we end up in negative remainder territory. We thus take 8 times 60 = 480.
541 -480 = 61.
8X4 = 32. 61 - 32 = 29.

Add the final number: ‘3’: 293
5X60 is certainly too big, so take 4x60 = 240. Leaves 53.
4X4=16. 53-16 = 37

We have used up all numbers from 235813 and the remainder is less than 64, so we can call out the answer:
3684 with a remainder of 37.

In this case I would use 40 as the divisor.
356 / 40. 9 seems ok since we will be adding 3 times 9 = 27 later.
9 times 40 = 360. R = minus 4.
Add 27 to get 23.

Add ‘0’: 230
230 / 40. 6 times 40 = 240. R = -10. 3 X 6 = 18. -10 + 18 = 8.

Add ‘9’.
89 / 40. 2 X 40 = 80., R = 9
9 + 3 X 2 = 15

Answer: = 962R15

Thanks Kinma and very well explained, it’s a great process but I’ll need to strengthen my memory muscles so I can keep track of everything, the example above with the negative remainder is challenging because you have to go back and “correct” your calculation. I’ll keep practising though.

In the example, in order to avoid the going back you could take 70 instead of 60. In that case you always subtract too much, so the correction is always an addition:

Let’s just do this:
235813 divided by 64. Use 70. Correction is now with a factor of 6 instead of 4.

235/70 = 3. 3X70 =210. 235-210 = 25
3X6 = 18. 25+18 = 43.

438/70 = 6. 6X70 = 420. 438-420 = 18
6X6 = 36. 18 +36 = 54

541/70 = 7. 7X70 = 490. 541-490 = 51
7X6 = 42. 51+42 = 93. Now we have an issue the other way round.
The remainder of 93 is bigger than 64. ‘7’ needs to become ‘8’.
The correction is easier now though. 93 - 64 = 29 and we use 8 instead of 7 in this step.
In other words: 7R93 = 8R29.

293/70 = 4. 4X70 = 280. 293 -280 = 13.
6X4 = 24. 13+24 = 37

Again, the answer is: 3684R37

Drawback is that when you have to divide by 61 and you use 70, the corrections become a lot greater than when you use 60.

I have talked about it a lot but imho the nine and eleven proof is great for strengthening those muscles.

Let’s do the eleven proof this time, and let’s do it in a way that keeps your brain calculating (there is a shorter way of doing the modulo 11).

So 235813 divided by 64 = 3684 R 37.
In other words: 3684 X 64 + 37 = 235813

We do a modulo 11 on all numbers by subtracting multiples of 11, so chosen that the numbers become smaller with each step:
3,684 - 3,300 = 384
384 - 330 = 54
54 - 44 = 10

64 - 55 = 9

37 - 33 = 4

We do the multiplication and addition again. However, now with the modulo 11 numbers:

10X9+4 = 94
Aain, do a modulo 11 on 94:
94 - 88 = 6

Now see if the answer also brings us back to 6:
235,813 - 220,000 = 15,813
15,813 - 11,000 = 4,813
4,813 - 4,400 = 413
413 - 330 = 83
83 - 77 = 6

6=6, so the answer is probably correct.

As always, it is a lot to type out.
If you get a little proficient in doing this, it is easier than it looks now.

9 proof using repeated subtraction (there is a way quicker method, but stick with me):

3,684 X 64 + 37 = 235,813

3,684 - 3,600 = 84
84 - 81 = 3

64 - 63 = 1

37 - 36 = 1

Do the calculation again:
3X1+1 = 4

Now the big one:
235,813 - 180,000 = 55,813
55,813 - 54,000 = 1,813
1,813 - 1,800 = 13
13 - 9 = 4

4 = 4, the calculation is correct.

By working so intensely with numbers you are sharpening your brain to work with bigger and bigger numbers.
In order to do one calculation you will be doing a lot of mental work and since you are going back to your original numbers a couple of times you will strengthen your memory for numbers.

Don’t try to work with big numbers too soon. Start small, get proficient, then move to bigger numbers.

Thanks again Kinma, makes sense.

Great! Just for fun I ordered the book.
There are not many books about the stuff you don’t learn in school.
I usually just find them, use them and then realise they in the middle ages they would use this kind of calculation.

This is very interesting, is there a mathematical proof that shows why this works?