Scale drawings and maps questions with answers

Questions

1. Question 1Map distance to actual distance

   A map is drawn to a scale of $1 : 100\,000$. Two towns are $2.4$ cm apart on the map. Find the actual distance, in kilometres, between the two towns.

   Show the worked solution

   1. Step 1: Write the scale as a ratio with units (1 km = 100 000 cm)$\mathstrut 1 : 100\,000$$\mathstrut 1\text{ cm} : 100\,000\text{ cm}$$1\text{ cm} : 1\text{ km}$
   2. Step 2: Multiply both parts by the map distance$\mathstrut 2.4\text{ cm} : 2.4 \times 1\text{ km}$$2.4\text{ cm} : 2.4\text{ km}$
2. Question 2Actual distance to map distance

   A map is drawn to a scale of $1 : 500\,000$. Two places are $36$ km apart. Find the distance, in centimetres, between the two places on the map.

   Show the worked solution

   1. Step 1: Write the scale as a ratio with units (1 km = 100 000 cm)$\mathstrut 1 : 500\,000$$\mathstrut 1\text{ cm} : 500\,000\text{ cm}$$1\text{ cm} : 5\text{ km}$
   2. Step 2: Scale up to the actual distance$\mathstrut \dfrac{36}{5}\text{ cm} : 36\text{ km}$$7.2\text{ cm} : 36\text{ km}$
3. Question 3Write a scale as 1 : n

   On a map, $4$ cm represents $5$ km. Express the scale of the map in the form $1 : n$.

   Show the worked solution

   1. Step 1: Write map : actual with units$\mathstrut 4\text{ cm} : 5\text{ km}$
   2. Step 2: Make both units cm (1 km = 100 000 cm)$4\text{ cm} : 500\,000\text{ cm}$That is $4 : 500\,000$.
   3. Step 3: Make the first part 1$\mathstrut 1 : \dfrac{500\,000}{4}$$1 : 125\,000$
4. Question 4Find the scale from two distances

   Two towns are $4$ km apart. On a map they are $5$ cm apart. Express the scale of the map in the form $1 : n$.

   Show the worked solution

   1. Step 1: Write map : actual with units$\mathstrut 5\text{ cm} : 4\text{ km}$
   2. Step 2: Make both units cm (1 km = 100 000 cm)$5\text{ cm} : 400\,000\text{ cm}$That is $5 : 400\,000$.
   3. Step 3: Make the first part 1$\mathstrut 1 : \dfrac{400\,000}{5}$$1 : 80\,000$
5. Question 5Scale drawings and plans

   The plan of a house is drawn to a scale of $1 : 200$. A room is $4.8$ cm long on the plan. Find the actual length, in metres, of the room.

   Show the worked solution

   1. Step 1: Write the scale as a ratio with units (1 m = 100 cm)$\mathstrut 1 : 200$$\mathstrut 1\text{ cm} : 200\text{ cm}$$1\text{ cm} : 2\text{ m}$
   2. Step 2: Multiply both parts by the length on the plan$\mathstrut 4.8\text{ cm} : 4.8 \times 2\text{ m}$$4.8\text{ cm} : 9.6\text{ m}$
6. Question 6Areas on maps

   A map is drawn to a scale of $1 : 200\,000$. A park has an area of $8.4$ cm² on the map. Find the actual area of the park, in km².

   Show the worked solution

   1. Step 1: Write the scale as a ratio with units (1 km = 100 000 cm)$\mathstrut 1 : 200\,000$$\mathstrut 1\text{ cm} : 200\,000\text{ cm}$$1\text{ cm} : 2\text{ km}$
   2. Step 2: Areas: square both parts of the length scale$\mathstrut 1^2\text{ cm}^2 : 2^2\text{ km}^2$$1\text{ cm}^2 : 4\text{ km}^2$
   3. Step 3: Multiply both parts by 8.4$\mathstrut 8.4\text{ cm}^2 : 8.4 \times 4\text{ km}^2$$8.4\text{ cm}^2 : 33.6\text{ km}^2$
7. Question 7A map distance, then the speed

   A map is drawn to a scale of $1 : 50\,000$. On the map, the route from a school to a sports centre is $32$ cm long. A cyclist leaves the school at $1655$ and must reach the sports centre at $1835$. Find the average speed, in km/h, at which the cyclist must travel.

   Show the worked solution

   1. Step 1: Write the scale as a ratio with units (1 km = 100 000 cm)$\mathstrut 1\text{ cm} : 50\,000\text{ cm}$$1\text{ cm} : 0.5\text{ km}$
   2. Step 2: The actual distance: multiply both parts by the map distance$32\text{ cm} : 16\text{ km}$
   3. Step 3: The time from 1655 to 1835, in hours (minutes ÷ 60)$100$ minutes $= \dfrac{100}{60}$ h
   4. Step 4: Average speed = distance ÷ time$\dfrac{16}{100 \div 60}$$\text{Speed} = 9.6$ km/h