For the reaction $\mathrm{A}\to \mathrm{B}$,  the rate of reaction becomes three times when the concentration of A is increased by nine times. What is the order of reaction ?

$\mathrm{A}\to \mathrm{B}$

Rate of reaction $r\propto [\mathrm{A}{]}^n$      …..(i)

If concentration of A is increased by nine times, then rate of reaction becomes three times,

$\begin{array}{l}{\mathrm{r}}^{\mathrm{\prime }}=3\mathrm{r}\\ {\mathrm{A}}^{\mathrm{\prime }}=9\mathrm{A}\\ {\mathrm{r}}^{\mathrm{\prime }}\propto {\left[{\mathrm{A}}^{\mathrm{\prime }}\right]}^{\mathrm{n}}\\ 3{\mathrm{r}}^{\mathrm{\prime }}\propto [9\mathrm{A}{]}^{\mathrm{n}}\end{array}$         …. (ii)

From eq. (i) and (ii) 

$\begin{array}{l}\frac{\mathrm{r}}{3\mathrm{r}}=\frac{[\mathrm{A}{]}^{\mathrm{n}}}{[9\mathrm{A}{]}^{\mathrm{n}}}\\ \frac{1}{3}={\left(\frac{1}{3}\right)}^{2\mathrm{n}}\\ 1=2\mathrm{n}\\ \mathrm{n}=1/2\end{array}$

$\therefore \text{ Order of reaction }=[1/2]$