Three digit division

Three digit division.
For most people this is a challenge.

Sometimes there are shortcuts, like the ones we discussed here:

In other cases you need a general way of working that is suitable for mental calculation.

I present here my general way of working.
It involves mostly 2 digit subtractions with each step. Or 3 digits where the 3rd digit is a zero. So each step should be easy to do.
The hard part - as always - is focusing on the numbers and keeping the numbers in your head.

An example will make this clear:

\frac{2359}{767}

First I mentally focus on 2359 and make a guess how many times 767 goes into 2359. Let’s try 3 times. The process is that we subtract first 3x700 from 2359, then 3X60 and then 3X7:

2359 - 2100 = 259. I focus on just the 2300 part of 59. So 2300 - 2100 = 200. This is a two digit subtraction.
Then I mentally add 59. Concatenate might be a better word for the mental process.
259 - 180 = 79. I subtract 200 from 200 to get 0 then add 20, add the 59 = 79.
79 - 21 = 58. Or do 260-180 = 80. 80-1 = 79.

Answer so far: 3 r 58.

Shift decimal point: 58 => 580.
767 does not go into 580.

Answer so far: 3.0 r 580

580 => 5800. 767 goes about 7 times into it.
5800 - 4900 = 900. Forget the zeros and just focus on the hundreds: 58 - 49.
900 - 420 = 480.
480 - 49 = 431. (480-50+1)

Answer so far: 3.07 r 431

431 => 4310. Six times is too much. Try 5 times:
4310 - 3500 = 810. Focus on 43 - 35, then add 10.
810 - 300 = 510.
510 - 35 = 475. I do 510 - 10 - 25.

Answer so far: 3.075 r 475

475 => 4750. 6 times?
4750 - 4200 = 550.
550 - 360 = 190.
190 - 42 = 148.

Answer so far: 3.0756 r 148

148 => 1480. 2 times?
1480 - 1400 = 80.
80 - 120 = -40.
-40 - 14 = -54.

Answer so far: 3.07562 r -54

-54 => -540. Zero times.

Answer so far: 3.075620 r -540

-540 => -5400. 7 times?
-5400 + 4900 = -500
-500 + 420 = -80
-80 + 49 = -31

From the answer - without digital separator - of 30756200 I mentally see or focus on only ‘200’.
I just subtract 7 from 200 = 193:

Answer so far: 3.0756193 r -31

-31 => -310. zero times.

Answer so far: 3.07561930 r -310

-310 => -3100. 4 times?
-3100 + 2800 = -300.
-300 + 240 = -60.
-60 + 28 = -32.

From the answer - without digital separator - of 3075619300 I mentally see and do is on only ‘300’ I just subtract 4 from 300 = 296:

Let’s stop here since the remainder is small.
Answer : 3.075619296 r -32

2 Likes

Great write-up Kinma! I like the self-correcting/adjusting that goes on in your method.

3 digit division is great, more challenging than 2 digit division yet still mentally do-able. I do 3 digit division the same way I do 2 digit division. So when I do 2359÷767, I divide 2359 by 767, which is 3 with a remainder of 58. I make 767 ten times smaller so 76,7 but this is still too large thus I make it again ten times smaller, 7,67 and divide 58 by 7,67 which is 7 with a remainder of 4,31. I can continue just like that.

What do you think of 4 digit division and 5 digit division? Have you ever done them before? I have never done 5 digit division before which makes it exciting :slight_smile: ohh, I can feel the numbers already.

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I don’t think I have done 4 digits. If it happens that I need to do one, I’ll probably round the numbers first.
But you are right; great challenge.