Shireen draws a circle in her courtyard. She then measures the circle’s circumference and its diameter with her measuring tape and records them as two integers, A and B respectively. She finds that A and B are co-primes, that is, their greatest common divisor is 1. She also finds their ratio, A:B, to be: 3.141614161416… (repeating endlessly).

What is A - B ?

We have 
3.1416141614161416......
=3+0.1416+0.00001416+0.000000001416.....
= 3+$$\frac{1416}{10^4}+\frac{1416}{10^8}+\frac{1416}{10^{12}}+......$$
Now excluding 3 we get a series with infinite Geometric progression such that first term is $$\frac{1416}{10^4}$$ and common ratio is $$\frac{1}{10^4}$$
Therefore we get sum as $$\frac{a}{1-r}$$
=$$\frac{1416}{\frac{10^4}{1-\frac{1}{10^4}}}$$
we get sum as $$\frac{1416}{10^4-1}$$
=$$\frac{1416}{9999}$$
Now adding 3 we get value as :
$$\frac{31413}{9999}$$
Now taking 3 common we  get ratio of A:B as
$$\frac{10471}{3333}$$
So A-B will be = 7138