A wire fixed at both its ends is found to vibrate at resonant...

For a wire fixed at both ends, the resonant frequencies are given by $f_n = n f_0$, where $n$ is an integer and $f_0$ is the fundamental frequency.

The given resonant frequencies are $240 \text{ Hz}$ and $320 \text{ Hz}$.

Let $n_1 f_0 = 240$ and $n_2 f_0 = 320$ for some integers $n_1$ and $n_2$.

This implies that $f_0$ must be a common divisor of $240$ and $320$.

The maximum value of the fundamental frequency is the greatest common divisor (GCD)$of 240$ and $320$.

$\text{GCD}(240, 320) = 80$

Thus, the maximum possible fundamental frequency is $80 \text{ Hz}$.