Squares in squares

Each of the "units" is similar, hence we only need to find the fraction in one of the units. Thus the fraction of pink area is simply:

$\dfrac{4}{12} = \dfrac{1}{3} \quad \implies \quad 1 + 3 = \boxed{4}$

This pattern is repeated:

In each part of the square, $4$ of $12$ triangles are shaded. This goes on infinitely, so the shaded area will be $\dfrac{4}{12} =\dfrac{1}{3}\implies a=1,b=3\implies \boxed{a+b=4}$

Let the length of each side of the square be $a$ . Then area of the square is $a^2$ . Total shaded area is the sum of the infinite G. P. series with first term $\dfrac {a^2}{4}$ and common ratio $\dfrac {1}{4}$

The total area of the shaded region is thus

$\dfrac {a^2}{4}\times \dfrac {1}{1-\frac{1}{4}}=\dfrac {a^2}{3}$ ,

and the ratio of the areas required is

$\dfrac {1}{3}$ . So, $a=1,b=3$ , and $a+b=\boxed 4$ .