A sum of amounts will be doubled in $7\frac{1}{2}$ years. What is the rate per cent?

Let the principal be $P$. Since the sum doubles itself, the final amount $A=2P$.
Therefore, the Simple Interest ($\text{SI}$) is:

$$\text{SI}=A-P=2P-P=P$$

The time period $T=7\frac{1}{2}\text{ years}=\frac{15}{2}\text{ years}$.
Using the Simple Interest formula:

$$\text{SI}=\frac{P\times R\times T}{100}$$

$$P=\frac{P\times R\times \frac{15}{2}}{100}$$

$$1=\frac{15R}{200}⟹R=\frac{200}{15}=\frac{40}{3}\%=13\frac{1}{3}\%\text{ per annum}$$