A certain sum of money lent out at simple interest amounts $₹13,000$ in $3\text{ years}$ and $₹15,000$ in $5\text{ years}$. Find the rate per cent per annum.

Let the principal amount be $P$ and the simple interest per year be $I$.According to the given information:
$$\text{Amount in }5\text{ years}=P+5I=15,000$$
$$\text{Amount in }3\text{ years}=P+3I=13,000$$
Subtracting the second equation from the first equation:
$$(P+5I)-(P+3I)=15,000-13,000$$
$$2I=2,000$$
$$I=1,000\text{ per year}$$
Now, find the principal $P$:
$$P+3(1,000)=13,000$$
$$P+3,000=13,000$$
$$P=10,000$$
The formula for simple interest for $1\text{ year}$ is:
$$I=\frac{P\times R\times 1}{100}$$
$$1,000=\frac{10,000\times R}{100}$$
$$1,000=100R$$
$$R=\frac{1,000}{100}=10\%$$
Thus, the rate of interest is $10\%\text{ per annum}$.