Arithmetic Techniques Part 2:
A way to approximate B/(A+d) with (Be)/A
Created by Sho Sengoku, 2002
In a backgammon game, especially in a match play
in a tournament, you often need to make a division something like 43/102,
whose denominator is close to a number easy to calculate, in this case
100. I'll explain how to approximate a fraction in that kind by shifting
its denominator to a number easy to calculate. You can calculate this type
of fractions mentally quite fast in most cases, once you get used to this
method.
When d is small enough comparing to A,
then by using e=B/A x d you can approximate
B/(A+d) with (Be)/A.

Description
Assume absolute value of d and e (d and e)
are small enough comparing to A.
Let C=B/A, d'=d/A, e'=e/A, then we get
B/(A+d) 
= C/(1+d') 
(1) 
(Be)/A 
= Ce' 
(2) 
Since we want to approximate B/(A+d) with (Be)/A,
by putting (1) = (2),
C/(1+d') 
=Ce' 

C 
=(Ce')(1+d') 


=C+Cd'e'e'd' 

Cd' 
=e'+e'd' 
(3) 
Since e' << 1, d' << 1,
e'd' is small enough to ignore
in (3),
Cd' 
~ e' 

e'/d' 
~ C 

e 
~ B/A x d 

Therefore, if e is chosen to be e=B/A x d,
B/(A+d) is approximated with (Be)/A.
Example application 1. 34/52
We are going to approximate 34/52 with something like r/100,
so we first multiply both denominator and numerator of the fraction with 2,
and get 68/104.
Then put d=4, and
e 
= 68/100 x d 


= 68 x 4 / 100 


= 272/100 


= 2.72 

Therefore you can approximate 34/52 with
34/52=68/104 
= 68/(100+4) 


~ (682.72)/100 


= 65.28/100 


= 0.6528 

Exact value of 34/52 is 0.65385, and approximation error is 0.00105.
Example application 2. 37/96
Put d=4 then e=37/100 x (4) = 148/100 = 1.48, so
37/96 
= 37/(1004) 


~ (37+1.48)/100 


= 38.48/100 


= 0.3848 

Exact value of 37/96 is 0.38542,
and approximation error is +0.00062.