Half-life periods for a reaction at initial concentration of 0.1M and 0.01M are 5 and 50 minutes respectively. Then the order of reaction is

When the initial concentration is reduced from 0.1 M to 0.01 M, the half-life period of the reaction increases from 5 minutes to 50 minutes.

Thus, when the initial concentration is reduced to one-tenth, the half-life period becomes 10 times.
Hence, the half-life period is inversely proportional to the initial concentration i.e. ${\mathrm{t}}_{1/2}\propto \frac{1}{\mathrm{A}}$

This is true for the second-order reaction.