Cubes form Cuboids

Let $\color{#D61F06}\large S_{cuboid}$ be the surface area of the cuboid and $\color{#D61F06}\large S_{cube}$ be the surface area of the cube.

$\color{#D61F06}\large S_{cuboid}=2(LW+WH+HL=2[3(1)+1(1)+1(3)=2(3+1+3)]=2(7)=14$

$\color{#D61F06}\large S_{cube}=6a^2=6(1^2)=6(1)=6$

$\color{#D61F06} \large ratio=\dfrac{6}{14}=\dfrac{3}{7}$

Finally,

$\color{#3D99F6} \large a+b=3+7=$ $\boxed{\color{#3D99F6} \large10}$

We know that the surface area of the cube $=6l^2$ .

Now, as we are joining three cubes to form a cuboid, we will have the following dimensions for the cuboid $3l\times l\times l$ . Hence the surface area of the cuboid will be: $2\times(3l\times l+l\times l+3l\times l)\\=2\times(7l^2)= \boxed{14l^2}$

Hence the ratio of cube:cuboid will be $\dfrac{6l^2}{14l^2}=\boxed{\dfrac 37}$ . Thus $a+b=3+7=\boxed{\boxed{\boxed{10}}}$ .

Formula for the surface area of the cuboid: $2(xz+yz+xy)$ where $x,y$ and $z$ are the edge lengths

Formula for the surface area of a cube: $6a^2$ where $a$ is the edge length of the cube

So the desired ratio is

$ratio=\dfrac{6(1^2)}{2[1(3)+1(1)+1(3)]}=\dfrac{6}{14}=\dfrac{3}{7}$

And the desired answer is $3+7=\boxed{10}$

Surface area of the cube: $S_{cube}=6a^2=6$

Surface area of the cuboid: $S_{cuboid}=2(LW+LH+HW)=2(3+3+1)=14$

$Ratio=6/14=3/7$

$a+b=3+7=10$

The length of the cube doesn't matter - call each face 1 unit; Each cube has 6 faces, so surface area --> 1 cube = 6 units in SA; 3 cubes have 18 units in SA. Since the cubes are joined, so that only one face touches to maximize, there are 4 faces total touching, which means there are 18-4=14 units of SA on the cuboid.

a/b = 6/14=3/7, 3+7=10