In order to find the ratio $\frac{a}{b}$, first express a and b as fractions.
$$a = 1+ \frac{\sqrt{3}}{3} = \frac{3+\sqrt{3}}{3}$$
$$b = 1 - \frac{\sqrt{3}}{3} = \frac{3-\sqrt{3}}{3}$$
Now, we take the ratio of a and b.
$$\frac{a}{b} = \frac{\frac{3+\sqrt{3}}{3}}{\frac{3-\sqrt{3}}{3}} = \frac{3+\sqrt{3}}{3-\sqrt{3}}$$
To get rid of the square root at the denominator, multiply by the conjugate.
$$\frac{a}{b} = \frac{3+\sqrt{3}}{3-\sqrt{3}} \times \frac{3+\sqrt{3}}{3+\sqrt{3}} = \frac{9+6\sqrt{3} + 3}{9-3} = \frac{12 + 6\sqrt{3}}{6}$$
Then you can simplify.