Let side AB = a and side AD = b
If we find the area of the shaded region then Area of Un-shaded region = Area of ABCD - area of the shaded region
So, lets find the area of the shaded region first.
Area of Shaded region = Area of the four triangles. Note that all 4 shaded triangles are congruent (Same dimensions)
So, Area of Shaded region = 4 * Area of one shaded triangle = 4 * Area of AEH
Let's find out area of AEH
AE = $\frac{1}{2}*AB$ = $\frac{a}{2}$ and AH = $\frac{b}{2}$
Area AEH = $\frac{1}{2}*AE*AH$ = $\frac{1}{2}*AE*AH$ = $\frac{1}{2}*\frac{a}{2}*\frac{b}{2}$ = $\frac{a*b}{8}$
=> Area of Shaded region = 4 * Area of AEH = $4*\frac{a*b}{8}$ = $\frac{a*b}{2}$
Area of Un-shaded region = Area of ABCD - area of the shaded region = a*b - $\frac{a*b}{2}$ = $\frac{a*b}{2}$
=> Area of Un-shaded region = area of the shaded region

So, ratio of the area of the shaded region to the area of the un-shaded region in the rectangle = 1:1

So, answer will be A
Hope it helps!