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The surface area of a cub is 6*(area of one face) = 6LW 
The volume of a cube is LWH 
acube = 6LW 

to make two equal cuboids, we cut the cube in half through its height - LW(.5H) 
The surface area of the cuboid is 
base + top + four sides 
LW + LW + 4(.5HL) 
2LW + 2LH 
acuboid = 2L(W+H) 

so the ratio of the cube to the cuboid is 
acube : acuboid 
6LW : 2L(W+H) 
now, the important part, since we started with a cube L = W = H, so we can simplify 
6LL : 2L(L+L) 
6L^2 : 2L(2L) 
6L^2 : 4 L^2 
2 4 2

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Let the side of solid cube = a
TSA of cube = 6a²

When the cube is cut into two cuboid with equal volume, the cube will be cut into two equal halves through its side.
the cuboid thus formed will have dimension,
length (l)= a
breadth (b)= a
height (h)= a/2
TSA = 2 (lb + bh + lh)
        = 2 (a² + a²/2 + a²/2)
        = 2 (2a²)
        = 4a²

Ratio = (TSA of cuboid) : (TSA of cube) = 4a² : 6a² = 2:3
The Ratio is 2:3.

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