Solution:

To find the time required for a particle in Simple Harmonic Motion (SHM) to travel from \( x = A \) to \( x = \frac{A}{2} \), we can use the following steps:

1. Angular Frequency (\( \omega \)):
\[
\omega = \frac{2\pi}{T}
\]

2. Displacement in SHM:
The position \( x(t) \) in SHM can be described by:
\[
x(t) = A \cos(\omega t)
\]

3. Finding Time for Positions:
- For \( x = A \):
\[
A \cos(\omega t_1) = A;  \cos(\omega t_1) = 1 ; t_1 = 0
\]

- For \( x = \frac{A}{2} \):
\[
\frac{A}{2} = A \cos(\omega t_2 ; \cos(\omega t_2) = \frac{1}{2} ; \omega t_2 = \frac{\pi}{3}
\]
\[
t_2 = \frac{\pi}{3\omega} = \frac{\pi T}{6}
\]

4. Time Interval:
The time taken to travel from \( x = A \) to \( x = \frac{A}{2} \) is:
\[
\Delta t = t_2 - t_1 = \frac{\pi T}{6} - 0 = \frac{T}{6}
\]

Thus, the time required to travel from \( x = A \) to \( x = \frac{A}{2} \) is:
\[{\frac{T}{6}}
\]