In two threads I showed how to make a division simpler by changing the divisor to a round number that is easier to work with:

Now let’s take a division by 23. We can either move 23 up to 25, or we can change it down to 20. Let’s do both and see which one is easier.

Btw., I consider 25 a ‘round’ number in terms of division. ‘Round’ being easy to use. Division by 25 is the same as multiplication by 4.

First moving up to 25.

The difference 25-23 is 2 so we correct the remainder by a factor of 2:

100/23

100/25 = 4r0 0+2*4=8
80/25 =3r5. 5+2*3=11

110/25 = 4r10. 10+2*4=18

180/25 = 7r5. 5+14 = 19

190/25 = 8r-10. -10+16 = 6

(if you started with 190/25=7 you quickly would have realized that the remainder gets too big: 190/25=7r15. 15+14 = 29. 29 > 25, so you have to use 8 instead of 7. It might feel strange at first (to use negative remainders) when you start to calculate this way. One way of dealing with this is to look ahead how much you need to add to the remainder and seeing if that does not become too big.)

60/25 =2r10. 10+4=14

140/25 = 6r-10. -10+12=2 (again 140/25=5 gives a remainder that is too big. So we have to use 6 instead of 5).

20/25 = 0r20

200/25 = 8r0. 0+16=16

160/25= 6r10. 10+12 = 22

220/25 = 9r-5 -5+18=13

130/25 = 5r5. 5+10=15

150/25 = 6r0…

Answer: 4,347826086956…

Compare this with rounding down.

We could round 23 down to 20 and do 100/20. Even better, we can also do 10/2 and correct 3 times the digit in the remainder.

Keep in mind that after we divide by 2, we need a remainder that is large enough to subtract 3 times the digit from.

100/23. Instead work with 10/2:

10/2 = 4r2. 20-3*4 =8 (10/2 of course is 5r0. However, this does not leave a remainder to subtract from. Therefore we take 4r2 instead of 5r0)
8/2 = 3r2. 20-3*3=11. (8 divided by 2 usually is 4, but again this leaves no remainder)

11/2 = 4r3. 30-12=18

18/2 = 7r4. 40-21=19

19/2 = 8r3. 30-24=6

6/2 = 2r2. 20-6=14

14/2= 6r2. 20-18=2

2/2=0r2. 20

20/2 = 8r4. 40-24 = 16

16/2 = 6r4. 40-18=22

22/2 = 9r4. 40-27=13

13/2= 5r3. 30-15=15

15/2= 6r3…

Answer again is: 4,347826086956…

With a bit of practice both ways can be executed quickly.

When rounding down, you need to ‘look ahead’ how much you need to subtract in order to have a remainder big enough.

When rounding up, you need to ‘look ahead’ how much you need to add in order to have a remainder small enough.

In the 18/2 part we just did you need to go down even two steps:

18/2 = 9r0. 0-3*9 is negative.
18/2 = 8r2. 20-3*8 is negative.

18/2 = 7r4. 40-3*7 is positive, so we need to use this one.