Sheet $\text{M}$ and sheet $\text{N}$ are of dimensions $6\:\text{cm} \times 4\: \text{cm}$ each.
On Sheets $M$ and $N$, if we perform the given operations, we get as follows.
Now, the perimeter of the final folded shape of sheet $M = 2(1.5+2) = 7\;\text{cm},$ and the perimeter of the final folded shape of sheet $N = 2(0.5+6) = 13\;\text{cm}.$
$\therefore$ The ratio of perimeters of the final folded shape of sheet $N$ to the final folded shape of sheet $M$ is $ = 13:7.$
So, the correct answer is $(A).$