Scale drawings and maps questions with answers

Questions

1. Question 1Map distance to actual distance

   A map is drawn to a scale of $1 : 50\,000$. Two towns are $5$ 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 : 50\,000$$\mathstrut 1\text{ cm} : 50\,000\text{ cm}$$1\text{ cm} : 0.5\text{ km}$
   2. Step 2: Multiply both parts by the map distance$\mathstrut 5\text{ cm} : 5 \times 0.5\text{ km}$$5\text{ cm} : 2.5\text{ km}$
2. Question 2Actual distance to map distance

   A map is drawn to a scale of $1 : 80\,000$. Two places are $6$ 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 : 80\,000$$\mathstrut 1\text{ cm} : 80\,000\text{ cm}$$1\text{ cm} : 0.8\text{ km}$
   2. Step 2: Scale up to the actual distance$\mathstrut \dfrac{6}{0.8}\text{ cm} : 6\text{ km}$$7.5\text{ cm} : 6\text{ km}$
3. Question 3Write a scale as 1 : n

   On a map, $4$ cm represents $2$ km. Express the scale of the map in the form $1 : n$.

   Show the worked solution

   1. Step 1: Write map : actual with units$\mathstrut 4\text{ cm} : 2\text{ km}$
   2. Step 2: Make both units cm (1 km = 100 000 cm)$4\text{ cm} : 200\,000\text{ cm}$That is $4 : 200\,000$.
   3. Step 3: Make the first part 1$\mathstrut 1 : \dfrac{200\,000}{4}$$1 : 50\,000$
4. Question 4Find the scale from two distances

   Two towns are $1.5$ km apart. On a map they are $6$ 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$\mathstrut 6\text{ cm} : 1.5\text{ km}$
   2. Step 2: Make both units cm (1 km = 100 000 cm)$6\text{ cm} : 150\,000\text{ cm}$That is $6 : 150\,000$.
   3. Step 3: Make the first part 1$\mathstrut 1 : \dfrac{150\,000}{6}$$1 : 25\,000$
5. Question 5Scale drawings and plans

   The plan of a house is drawn to a scale of $1 : 250$. A room is $6.2$ 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 : 250$$\mathstrut 1\text{ cm} : 250\text{ cm}$$1\text{ cm} : 2.5\text{ m}$
   2. Step 2: Multiply both parts by the length on the plan$\mathstrut 6.2\text{ cm} : 6.2 \times 2.5\text{ m}$$6.2\text{ cm} : 15.5\text{ m}$
6. Question 6Areas on maps

   On a map, an area of $1$ cm² represents an actual area of $9$ km². Two schools are $8$ cm apart on the map. Find the actual distance, in kilometres, between the two schools.

   Show the worked solution

   1. Step 1: Square-root both parts to get the length scale$\mathstrut 1\text{ cm}^2 : 9\text{ km}^2$$\mathstrut 1\text{ cm} : \sqrt{9}\text{ km}$$1\text{ cm} : 3\text{ km}$
   2. Step 2: Multiply both parts by 8$\mathstrut 8\text{ cm} : 8 \times 3\text{ km}$$8\text{ cm} : 24\text{ km}$
7. Question 7A map distance, then the speed

   A map is drawn to a scale of $1 : 250\,000$. On the map, the route from a school to a sports centre is $8.4$ cm long. A coach leaves the school at $0655$ and must reach the sports centre at $0725$. Find the average speed, in km/h, at which it 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} : 250\,000\text{ cm}$$1\text{ cm} : 2.5\text{ km}$
   2. Step 2: The actual distance: multiply both parts by the map distance$8.4\text{ cm} : 21\text{ km}$
   3. Step 3: The time from 0655 to 0725, in hours (minutes ÷ 60)$30$ minutes $= \dfrac{30}{60}$ h
   4. Step 4: Average speed = distance ÷ time$\dfrac{21}{30 \div 60}$$\text{Speed} = 42$ km/h