 # Propose of a relaxing division method

All of us know that division is mainly about subtraction, but can we do it in a relaxing way. The answer is affirmative!

Assume that a division question does not have a remainder. Questions to be analysed: 4-digit divides 2-digit.
divisor x quotient = dividend, p x q = r, e.g. 13x65=0845

1. Find the tens-digit
</py code>

# round the dividend to nearest 100

``````r2 = round(r, -2)
``````

# round the divisor to 0/5

``````if p%10 in [0, 1, 2]:
p2 = p//10*10
elif p%10 in [3, 4, 5, 6, 7]:
p2 = p//10*10+5
else:
p2 = p//10*10+10
print(r2//10, '/', p2)
q2 = math.floor(r2/p2)
``````
1. Find the units-digit

### 9x9 multiplication table modolus of 10

This is just the traditional multiplication table we learnt, so no worries!
Just look at the units-digit as 4x9=36 → 6
1 2 3 4 5 6 7 8 9
2 4 6 8 0 2 4 6 8
3 6 9 2 5 8 1 4 7
4 8 2 6 0 4 8 2 6
5 0 5 0 5 0 5 0 5
6 2 8 4 0 6 2 8 4
7 4 1 8 5 2 9 6 3
8 6 4 2 0 8 6 4 2
9 8 7 6 5 4 3 2 1
e.g. 1472 / 16 = 92
left column = 6 (from divisor 16)
element = 2 (from dividend 1472)
top row = 2/7 (from quotient 92)

example 1 :
50 x 37 = 1850
1850 / 50 = ?
1800 / 50
36
final decision: 36 → 37 bc 1850>1800

example 2 :
67 x 44 = 2948
2948 / 67 = ?
2900 / 65
44

final decision: 4 4

example 3 :
76 x 63 = 4788
4788 / 76 = ?
4800 / 75
64
[3, 8] //4 is closer to 3 than 8
final decision: 6 3

example 4 :
89 x 52 = 4628
4628 / 89 = ?
4600 / 90
51

final decision: 5 2

example 5 :
32 x 49 = 1568
1568 / 32 = ?
1600 / 30
53
[4, 9] // 3 is closer to 4 than 9
final decision: 5 4

example 6 :
56 x 34 = 1904
1904 / 56 = ?
1900 / 55
34
[4, 9]
final decision: 3 4

example 7 :
45 x 87 = 3915
3915 / 45 = ?
3900 / 45
86
[1, 3, 5, 7, 9] // this is what makes this method so-called relaxing game final decision: 8 5 → 87 because 3915>3900

example 8 :
65 x 72 = 4680
4680 / 65 = ?
4700 / 65
72
final decision: 72

example 9 :
11 x 54 = 594
594 / 11 = ?
600 / 10
60

final decision: 6 4

example 10 :
73 x 62 = 4526
4526 / 73 = ?
4500 / 75
60

final decision: 6 2

To conclude, it is a quick approximation to get a division answer for the Non-remainder type question.

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Nice—and I guess what makes it “relaxing” is that it doesn’t use much working memory at any instant.

This technique is used extensively by the fastest people in the Memoriad Division category. However it doesn’t work when the divisor ends in 5: e.g. 51 x 55 = 2805, as any number ending in 5 multiplied by an odd number gives a answer ending in 5.

Note that at least one of your examples is incorrect (example 9) as the rounding 11 → 10 is too crude. But this works most of the time and is easier than full division.

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Very nice explanation @Antelex

Daniel is exactly right. Speak for my own experience (generally) both the divisor and dividend end with 5 are the most difficult ones, end with even numbers are the second, end with one or two odd numbers are the easiest.
The time spent are roughly 2:1.5:1.(for 6/3, as the digits goes up the gap will narrow)

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Thanks for all the replies, I appreciate that @Daniel_360 you watched thoroughly. You are a careful teacher! And @flou my training is closed to your experience time. Cool!

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