A guitar string of length 90cm vibrates with a fundamental...

The formula to calculate the fundamental frequency produced in a vibrating string is given by

${f}_{0} = \dfrac{v}{2L} . . . \left(1\right)$

For the first case, it can be written that

$120 = \dfrac{v}{2 ( 90 )} . . . \left(2\right)$

Similarly, for the second case, it can be written that

$180 = \dfrac{v}{2 ( L )} . . . \left(3\right)$

Divide equation (3) by equation (2) and solve to obtain the required length of the string.

$\begin{cases}\dfrac{180}{120} & = & \dfrac{\dfrac{v}{2\left(L\right)}}{\dfrac{v}{2\left(90\right)}} \\ & = & \dfrac{90}{L} \\ & ⇒ & L = 90 \times \dfrac{120}{180} \\ & = & 60\end{cases}$