There is a simple trick to quickly do this.
As an example let’s take 100/59.
In order to calculate this quickly., instead of 100/59, we do 10/6 with a small and simple correction at each step. We add - concatenate might the better word - the digit found to the remainder.
Instead of 100/59, we start with 10/6:
10/6=1r4. Concatenate the ‘1’ to the remainder. So 1r4 becomes 1r41.
41/6=6r5 => 6r56
56/6=9r2 => 9r29
29/6=4r5 => 4r54
54/6=9r0 => 9r09
9/6=1r3 => 1r31
31/6=5r1 => 5r15
15/6=2r3 => 32
32/6=5r2 => 5r25
25/6=4r1 => 14
14/6=2r2 => 22
22/6=3r4 => 43
43/6=7r1 => 17
17/6=2r5 => 52
52/6=8r4 => 48
48/6=8r0 => 08
8/6=1r2 => 21
Answer: 0.16949152542372881
We do the same thing for numbers ending in 8.
Now we add twice the digit to the remainder.
100/58
10/6 = 1r4 => 42
42/6 = 7r0 => 14
14/6 = 2r2 => 24
24/6 = 4r0 => 8
8/6 = 1r2 =>22
22/6 = 3r4 =>46
46/6 = 7r4 => 54 (40+14)
54/6 = 9r0 => 18
18/6 = 3r0 => 06
etc.
Answer: 1.72413793