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 same rate of simple interest. Find the sum.

Let the Principal amount be $P$ and the annual simple interest be $I$.
The total amount after $10$ years is given by:

$$A_{10}=P+10I=1600$$

The total amount after $4$ years is given by:

$$A_4=P+4I=1240$$

Subtracting the second equation from the first to isolate the interest earned over the $6$ -year interval:

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

$$6I=360⟹I=\frac{360}{6}=60$$

So, the interest earned per single year is $\text{Rs.}60$.
Now, calculate the cumulative simple interest earned over $4$ years:

$$\text{SI for }4\text{ years}=4\times 60=240$$

Subtract this interest from the $4$ -year amount to find the initial Principal ($P$):

$$P=A_4-240=1240-240=1000$$

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