(a) If ${t_1}$ and $2{t_2}$ are the time taken by particle to cover first and second half distance respectively.

${t_1} = \frac{{x/2}}{3} = \frac{x}{6}$…(i)

${x_1} = 4.5\;{t_2}$ and ${x_2} = 7.5\;{t_2}$

So, ${x_1} + {x_2} = \frac{x}{2} \Rightarrow 4.5{t_2} + 7.5{t_2} = \frac{x}{2}$

${t_2} = \frac{x}{{24}}$…(ii)

Total time $t = {t_1} + 2{t_2} = \frac{x}{6} + \frac{x}{{12}} = \frac{x}{4}$

So, average speed $ = 4\;m/sec$.