factoring out $10^2$ and $10^3$ and than reducing the fraction by $10^2$ is one way to deal with this question:

$\frac{10^8-10^2}{10^7-10^3}=\frac{10^2(10^6-1)}{10^3(10^4-1)}=\frac{10^6-1}{10*(10^4-1)}$.

Now, $10^6-1$ is very close to $10^6$ and $10^4-1$ is very close to $10^4$, hence $\frac{10^6-1}{10*(10^4-1)}\approx{\frac{10^6}{10*10^4}=\frac{10^6}{10^5}=10}$.

Answer: B.

Or else you can notice that we need approximate value of a fraction. Now, $10^{8}$ is much, much, much bigger than $10^{2}$. So subtracting $10^{2}$ from $10^{8}$ will be very close to $10^{8}$, basically $10^{2}$ is negligible in this case. The same for for $10^{7}$ and $10^{3}$. So $\frac{10^8-10^2}{10^7-10^3} \approx{\frac{10^8}{10^7}}=10$.

Hope it helps.