1. **State the problem:** We have a map scale of 1:10,000, meaning 1 cm on the map represents 10,000 cm in reality. 2. **Part (e): Find the actual distance represented by 8.2 cm on the map.** Formula: Actual distance = Map distance $ \times $ Scale factor 3. Calculate: $$\text{Actual distance} = 8.2 \times 10,000 = 82,000 \text{ cm}$$ Convert to metres: $$82,000 \text{ cm} = \frac{82,000}{100} = 820 \text{ m}$$ 4. **Part (a): Find the map distance representing an actual distance of 3 km.** Convert 3 km to cm: $$3 \text{ km} = 3,000 \text{ m} = 300,000 \text{ cm}$$ 5. Use the proportion: $$\frac{1}{10,000} = \frac{x}{300,000}$$ Cross multiply: $$10,000x = 300,000$$ Divide both sides by 10,000: $$\cancel{10,000}x = \frac{300,000}{\cancel{10,000}}$$ $$x = 30 \text{ cm}$$ 6. **Final answers:** - Actual distance for 8.2 cm on map: **820 metres** - Map distance for 3 km actual: **30 centimetres**