If the angles of a triangle are in the ratio $3:5:10$ then the ratio of smallest side to greatest side is

Given $A:B:C=3:5:10$

Here $A$ is smallest angles and $C$ is longest angles

$A+B+C=180°$

So that $A=30°,C=100°$

Hence, the opposite sides are smalles and longest

$\begin{array}{rcl}\frac{a}{c}& =& \frac{2R\sin A}{2R\sin B}\\ & =& \frac{\sin 30°}{\sin 100°}\\ & =& \frac{1}{2\cos 10°}\end{array}$

Therefore, the ratio of shortest and longest side is $\boxed{1:2\cos 10°}$