At the same rate of simple interest per annum, if a principal becomes the amount of Rs. $7100$ in $7$ years and of Rs. $6200$ in $4$ years, determine the principal amount.

Let the principal be $P$ and the simple interest per year be $I$.
Amount in $7$ years: $P+7I=7100$
Amount in $4$ years: $P+4I=6200$

Subtracting the second equation from the first:

$$(P+7I)-(P+4I)=7100-6200$$

$$3I=900⟹I=\text{Rs. }300\text{ per year}$$

Now substitute the value of $I$ into the equation for $4$ years:

$$P+4(300)=6200$$

$$P+1200=6200⟹P=6200-1200=\text{Rs. }5000$$