Please help with maths question?

The thing to remember is that, for similar shapes, while linear distances vary on a linear scale, areas vary on a squared scale (and volume varies on a cubed scale). In other words, for two shapes which are similar but of different sizes, we can say that given a linear scaling factor k, the scaling factor for the areas is k².

i) The linear scaling factor is 4 km/cm, so the area scaling factor is 16 km²/cm²:

(8 km²) / (16 km²/cm²) = 0.5 cm².

ii) The linear scaling factor is 50,000 km/km, so the area scaling factor is 50,000² km²/km²:

(8 km²) / (50,000² km²/km²) = 3.2x10^-9 km².

This is correct, but the use of the unit km² is cumbersome. Changing the the km² to cm²:

(3.2x10^-9 km²)(1000² m²/km²)(100² cm²/m²) = 32 cm².

Notice how the linear conversion factors were squared to get the area conversion factors, as mentioned above.