A sum of money kept in a bank amounts to $\text{Rs. }1,240$ in $4$ years and $\text{Rs. }1,600$ in $10$ years at the same rate of simple interest. Find the sum.

Let the Principal amount be $P$ and the simple interest per year be $I$.
According to the question:

$$P+10I=1600\text{— (Equation 1)}$$

$$P+4I=1240\text{— (Equation 2)}$$

Subtracting Equation 2 from Equation 1:

$$(P+10I)-(P+4I)=1600-1240$$

$$6I=360$$

$$I=\frac{360}{6}=\text{Rs. }60\text{ per year}$$

Now substitute the value of $I$ into Equation 2 to find $P$:

$$P+4(60)=1240$$

$$P+240=1240$$

$$P=1240-240=\text{Rs. }1,000$$

Thus, the initial sum (principal) is $\text{Rs. }1,000$.