If you have two tuning forks and you know that one tuning fork is 305 Hz, and when the tuning forks are pinged together you get a beat frequency of 6 Hz, then you know that the other tuning fork is either 311 Hz or 299 Hz. This idea I am fine with.
Now, the second part is as follows: if you replace the known tuning fork with one that emits a higher frequency, the beat frequency is 4 Hz. What is the frequency of the second tuning fork?
The answer is 311 Hz. But I don't understand how we can arrive there. The two things we know are $1.$ that the new frequency is $>$305 Hz and $2.$ that the beat frequency decreases from 6 Hz to 4 Hz. How does this then tie together that the other frequency is definitely 311 Hz?
The way it was explained was that there are two situations: if the other frequency was 306 Hz (why? where did 306 Hz come from?) then the beat frequency would increase and if the other frequency was 311 Hz (again why?) then the beat frequency would decrease. Since we know the beat frequency decreases from 6 Hz to 4 Hz, it must be the second case.