Work Step by Step
For constructive interference, the path difference equals $m \lambda$, where m = 0, 1, 2, 3, etc. Cutting the frequency in half will double the wavelength. The new wavelength $\lambda$ is twice as long as the original wavelength $\lambda_0$. Some students who heard a loud tone before were at locations for which the path difference was $\lambda_0, 3\lambda_0, 5\lambda_0$, etc. The new wavelength is $\lambda=2\lambda_0$, so these same students are now at a point where the path differences are $\lambda/2, 3\lambda/2, 5\lambda/2$, etc. These are now points of destructive interference, and they will hear nothing. On the other hand, for the students who heard a loud noise who were at points where the path difference was $2\lambda_0, 4\lambda_0, 6\lambda_0$, etc., the path difference is now $\lambda, 2\lambda, 3\lambda$, etc. So they will still hear a loud tone.