What is the first nonzero digit to the right of the decimal point of the fraction $\frac{1}{129}$?
I notice that 129 is close to 130, which is a multiple of 10. That means that $\frac{1}{129}$ is close to $\frac{1}{130}$, which is $\frac{1}{10} \cdot \frac{1}{13}$. That means that $\frac{1}{13}$ is a repeating decimal that starts with 0.076923. Therefore, $\frac{1}{10} \cdot \frac{1}{13}$ is the same decimal shifted one place to the right, so it starts with 0.0076923. This is very close to $\frac{1}{129}$, so the first nonzero digit to the right of the decimal point of $\frac{1}{129}$ must be the same as the first nonzero digit of $\frac{1}{10} \cdot \frac{1}{13}$, which is 7. 