Orange Slices

Let $r$ be the radius of the orange. Thus, originally the surface area of the sphere = $4\pi r^2$ .

After cutting, each slide is semi-circular in shape with the same radius, and since there are two sides of the slice, the additional surface area is double of the semi-circle or one full circle area. Thus, each slide has additional area of $\pi r^2$ .

As a result, all the 8 slices have the surface area = $4\pi r^2$ + $8\pi r^2$ = $12\pi r^2$ .

Hence, the surface has increased by $\boxed{3}$ -fold.

Surface area of one slice $=$ $\frac{ɸ}{360}$ $(4πr^2) + πr^2 =$ $\frac{ɸ}{90}$ $πr^2+ πr^2$

Since there are 8 equal slices $=$ $ɸ = 360/8 = 45$ $degrees$

Now substitute the value of $ɸ$ , we have

Surface area of one slice $=$ $\frac{45}{90}$ $(πr^2+ πr^2) = 1.5 πr^2$

To obtain the total surface area of the eight equal slices, we need to multiply the result above by eight, we have

Surface area of eight slices $= 8(1.5 πr^2) = 12 πr^2$

ratio of the total surface area of the 8 slices to that of the original orange $= (12 πr^2)/(4πr^2) = 3/1$

Let r be the radius of the orange.

Surface Area of the whole orange is (4)(pi)(r^2)

Surface Area of the 8 slices is the surface area of the orange (the rind stays the same) + the surface area of the fleshy parts. If you imagine the orange cut in half, it is easy to see that the fleshy part is a circle with radius r, as those halves are cut further, the fleshy parts become 1/2 circles of radius r. Each slice has 2 fleshy parts, or 1/2 circle + 1/2 circle = 1 circle. The area of that circle is (pi)(r^2); there are 8 slices, so the fleshy parts add up to 8(pi)(r^2).

Thus our ratio (Slices/Whole) is [4(pi)(r^2) + 8(pi)(r^2)] / 4(pi)(r^2) Factoring out the (pi)(r^2) leaves us with: (4+8)/4 = 12/4 = 3.

The answer is 3.

Observed that the surface area of one slice is equal to the area of the lune plus the area of one circle. So the total surface area of the $8$ slices is equal to the surface area of the spherical orange plus the area of $8$ circles. The desired ratio is therefore,

$\dfrac{4\pi r^2+8\pi r^2}{4\pi r^2}=\boxed{3}$ .