1. **Problem statement:** (a) Given the scale of a map is 1 : 250000, and the actual distance between two cities is 80 km, calculate the distance on the map in centimetres. 2. **Formula and explanation:** The scale means 1 cm on the map represents 250000 cm in reality. 3. **Convert actual distance to centimetres:** Since 1 km = 100000 cm, $$80\text{ km} = 80 \times 100000 = 8000000 \text{ cm}$$ 4. **Calculate map distance:** Using the scale, $$\text{Map distance} = \frac{\text{Actual distance}}{\text{Scale factor}} = \frac{8000000}{250000}$$ 5. **Simplify the fraction:** $$\frac{\cancel{8000000}}{\cancel{250000}} = 32$$ 6. **Answer for (a):** The distance on the map is **32 cm**. 1. **Problem statement:** (b) On the map, a forest area is 6 cm². Calculate the actual area in square kilometres. 2. **Formula and explanation:** Area scale factor is the square of the linear scale factor. 3. **Calculate actual area:** $$\text{Actual area} = \text{Map area} \times (\text{Scale factor})^2 = 6 \times (250000)^2$$ 4. **Calculate:** $$(250000)^2 = 62500000000$$ 5. **Multiply:** $$6 \times 62500000000 = 375000000000 \text{ cm}^2$$ 6. **Convert cm² to km²:** Since $1 \text{ km}^2 = (100000 \text{ cm})^2 = 10000000000 \text{ cm}^2$, $$\text{Actual area} = \frac{375000000000}{10000000000} = 37.5 \text{ km}^2$$ 7. **Answer for (b):** The actual area of the forest is **37.5 km²**.