Arithmetic Techniques Part 2: A way to approximate  B/(A+d) with (B-e)/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 (B-e)/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) (B-e)/A = C-e' (2)

Since we want to approximate B/(A+d) with (B-e)/A, by putting (1) = (2),
 C/(1+d') =C-e' C =(C-e')(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 (B-e)/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) ~ (68-2.72)/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/(100-4) ~ (37+1.48)/100 = 38.48/100 = 0.3848

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