The rate law expression is

$$\mathrm{R}=k[\mathrm{G}]^{a}[\mathrm{H}]^{b} $$ ...... (i)

The rate of reaction increases by a factor of 2 on doubling the concentration of '$$\mathrm{G}$$'.

The rate law expression becomes

$$\mathrm{R}^{\prime}=2 \mathrm{R}=k 2^{a}[\mathrm{G}]^{a}[\mathrm{H}]^{b}$$ ..... (ii)

Divided equation (ii) with eqn. (i)

$$\begin{aligned}\frac{\mathrm{R}^{\prime}}{\mathrm{R}} & =\frac{2 \mathrm{R}}{\mathrm{R}}=\frac{k 2^{a}[\mathrm{G}]^{a}[\mathrm{H}]^{b}}{k 2[\mathrm{G}]^{a}[\mathrm{H}]^{b}} \\2 & =2^{a} \\a & =1\end{aligned}$$

The rate of reaction increases by a factor of 8 on doubling the concentration of $$\mathrm{G}$$ and '$$\mathrm{H}$$'.

The rate law expression becomes

Rate $$=\mathrm{R}^{\prime \prime} k 2^{a}[\mathrm{G}]^{a} 2^{b}[\mathrm{H}]^{b}$$ ..... (iii)

Divide equation (iii) with equation (i)

$$\begin{aligned}\frac{8 \mathrm{R}}{\mathrm{R}} & =\frac{k 2^{a}[\mathrm{G}]^{a} 2^{b}[\mathrm{H}]^{b}}{k[\mathrm{G}]^{a}[\mathrm{H}]^{b}}\end{aligned}$$

$$8=2^a2^b$$

or $$b=2$$

The overall order of reaction is

$$a+b=1+2=3$$