The ratio of linear expansivity to the co-efficient of a real expansion of a rectangular sheet of a solid is

Let us suppose that the length of the rod is, $l$, then on the increasing temperature the relation between change in length and coefficient of linear expansion, $\alpha$ is, 
$∆l=l\left(1+\alpha ∆T\right)$ and the relationship between change in area and coefficient of superficial expansion, $\beta$, is, 
$∆A=A\left(1+\alpha ∆T\right)$, let us suppose that the length of the sides are one meter then, 

$∆l=1\times \left(1+\alpha ∆T\right)$ and $∆A=1\times \left(1+\beta ∆T\right)$
$$\begin{gathered} \Rightarrow ∆A=∆l\times ∆l=1\times \left(1+\beta ∆T\right)={\left(1+\alpha ∆T\right)}^2 \\ \Rightarrow \left(1+\beta ∆T\right)={\left(1+\alpha ∆T\right)}^2=\left(1+2\alpha ∆T\right)\Rightarrow \beta =2\alpha \\ \Rightarrow \frac{\alpha }{\beta }=0.5 \end{gathered}$$