What type of interference would occur between two frequencies that share the ratio 2:1? Let's say I have two sine wave oscillations of 200Hz and 100Hz playing at the same time - what type of interference will I observe and why? 
 A: When a signal is the superposition of two sinusoidal waves with a fixed phase difference the resulting signal in general can be described as a sinusoidal wave with a frequency which is the arithmetic average of the two original frequencies but with an amplitude which is varying with a frequency which is the semi-sum of those frequencies. So, in your case the resulting signal would behave as a signal at frequency $150$ Hz whose amplitude oscillates with frequency $50$ Hz  (frequency of beats).
Although the  math behind is connected to the properties of trigonometric functions, the simplest way to prove the result is by working with the complex representation of the time variation:
$$
y_1(t)=A_1 e^{i\omega_1 t} \\
y_2(t)=A_2 e^{i\omega_2 t}  \\
y(t)=y_1(t)+y_2(t)= A_1 e^{i\omega_1 t}+A_2 e^{i\omega_2 t} = \\=\left(A_1 e^{i \frac{ \left( \omega_1 - \omega_2 \right)}{2} t}+A_2 e^{-i \frac{ \left( \omega_1 - \omega_2 \right)}{2} t} \right) e^{i \frac{ \left( \omega_1 + \omega_2 \right)}{2} t}
$$
As far as the "why?", the answer is more difficult if you are looking for something more intuitive than just mathematical formulae. Pictures like the last in this web page may help in visualizing things but they do not go really deeper than the formulae.
