Four digit division with divisors slightly lower than a multiple of hundred

The base method is explained here:


This post explains how I do four digit division when the numbers can be easily rounded up.

4 digit division can be done with the same 2 step method.
If the divisor is close to a multiple of hundred, we can just take that number and add (multiples of) the difference to the remainder.

Lat’s try: 10,000 / 3,787
Mentally, split the 10,000 into 100 | 00 and 3787 into 37 | 87.
Round 37|87 to 38|00. Difference is 13.a
Also round the 38 to 40. Difference is 2.

1: Focus on the left parts: 100 and the 40. Try 2. 2 times 40 = 80 and 100 - 80 = 20. 20 + 2*2 (twice the difference) = 24. The 24 is actually 24|00.
2: We now subtracted too much. The difference is 13. So just add 2 times 13 = 26. 24|00 + 26 = 24|26 or 2426.

Add zero: 242|60.
1: Again just focus on 242 and 38. We rounded 38 to 40. We easily see 6 times. 6 times 40 makes 240. 242 - 240 = 2. Now go back from 40 to 38. Add 2 times 6 = 12. 2+12= 14. What we did was this 242 - 6*38 = 14. Combine with the ‘|60’ gets 1460.
2: We subtracted too much. Add 6 times (10 + 3): 1460 + 60 +18. 1520 + 18 = 1538.

Add zero. 153|80.
1: Try 4 times. Again round 38 to 40. 153-160 = -7. -7 + 24 = 1.
2: 1|80 = 180 + 4
13 = 180 + 40 + 12 = 220+12 = 232.

Add zero: 2320. Lower than 3738, so digit is zero.

Add zero: 232|00.
1: 232 / 40 suggests 5 times. However 6 times 40 is 240 and very close to 232. 232 - 240 = -8. Add 6 times 2 (difference 40-38) and get 4.
2: 4|00 + 6 times 13 = 400 + 60 + 18 = 478.

Add zero. 47|80.
1: Digit is one. 47 - 40 + 2 = 9.Combine => 9|80
2: 980 +13 = 993.

Add zero. 99|30.
1: 2 times. 99 - 240 = 19. 19 + 22 = 23.
2: 23|30. Leave the 23 and focus on the 30. We now need to add 2*13 = 26. 30+26 = 56. Combine: 23|56 = 2356.

Add zero. 235|60.
1: Remember that we did 232 instead of 235 3 steps ago. So 6 times.
235 - 640 = -5. -5 + 26 = -5 +12 = 7.
2: 7|60 + 613 = 8|20 + 63 = 838.

Next digit shall be 2 (because 83 goes twice into 40).

Answer so far: 2.64061262.

Observe:

  • That we split the divisor and remainder into 2 parts. One that we work with and one part that we just leave be.
  • We round up for easy division. 38 becomes 40. Adjust for the difference afterwards.
  • In no step have we multiplied with 3787 or even 3800. We just take 40, then account for the difference.
  • Each step is a division of maximum 3 digits with maximum 2. And even this, we mentally do 2 digits divided by 1. Because even though the number 40 consists of 2 digits, a division of a 3 digit number by 40, is actually a two digit divided by one digit. Mentally, if we do 235 /40, we focus on 23 / 4.
  • Each step therefore is (very) easy to do.
  • This method also makes sure the remainder that we work with gets smaller with each step (except for the each time we add a zero to the remainder). 242|60 becomes 14|60 (one digit less to remember).
  • We work with only 3 digits at the same time. The rest just stays the same.
  • A division with a divisor that we can round up - as in the example above - is actually easier to work with; because we add a difference at each step.
  • The remainder is 5 digits max. These you have to keep in your short time memory. You split these into 2 parts: 3 digits and 2. When working with the first 3 digits, you can (temporarily) forget about the other 2 (until you need them in the next step).

Try this for a while. It is easier than it seams.
The steps are easy and short.

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