A sum becomes $\text{Rs. }13,520$ in $2$ years at $4\%$ per annum compound interest. Find the original sum.

The formula for the amount under compound interest is:
$$A=P{\left(1+\frac{R}{100}\right)}^T$$

Given:

* Amount ($A$) = $\text{Rs. }13,520$
* Rate ($R$) = $4\%$ per annum
* Time ($T$) = $2$ years

Substituting the values into the formula:

$$13,520=P{\left(1+\frac{4}{100}\right)}^2$$

$$13,520=P{\left(1+\frac{1}{25}\right)}^2$$

$$13,520=P{\left(\frac{26}{25}\right)}^2$$

$$13,520=P\times \frac{676}{625}$$

Solving for $P$:

$$P=\frac{13,520\times 625}{676}$$

Since $676\times 20=13,520$:

$$P=20\times 625=12,500$$

Thus, the original sum is $\text{Rs. }12,500$.