Question: A length of $5$ cm on a map represents an actual distance of $1$ km. (a) Find the scale of the map in the form $1 : n$. (b) A path is $36$ cm on the map — find its actual length in metres. (c) A compound's perimeter is $1.3$ km — find its length on the map in cm.

Step 1. (a) Same units first, then reduce the left side to 1:

$$5 \text{ cm} : 1 \text{ km} = 5 \text{ cm} : 100\,000 \text{ cm} = 1 : 20\,000$$

Step 2. Keep the working form "$1$ cm : $20\,000$ cm", i.e. $1$ cm on the map $= 200$ m on the ground.

Step 3. (b) Map → ground, multiply:

$$36 \text{ cm} \times 200 \text{ m/cm} = 7200 \text{ m}$$

Step 4. (c) Ground → map, divide:

$$1.3 \text{ km} = 1300 \text{ m} \;\Rightarrow\; \dfrac{1300}{200} = 6.5 \text{ cm}$$

Check. Bigger real distances need more map centimetres: $1.3$ km $>$ $1$ km should give more than $5$ cm — and $6.5 > 5$. ✓

⚠ Watch out: A scale $1 : n$ has no units — both sides are the same unit, so $1$ km must become $100\,000$ cm before dividing. Keep the directions straight: map→ground multiplies by the scale factor, ground→map divides.

Question: The scale of a map is $1 : 40\,000$. (a) Find, in km², the actual area represented by $125$ cm² on the map. (b) Find, in cm², the map area representing an actual $4.8$ km².

Step 1. Convert the distance scale into friendly units:

$$1 \text{ cm} : 40\,000 \text{ cm} = 1 \text{ cm} : 0.4 \text{ km}$$

Step 2. Square both sides to get the area scale:

$$(1 \text{ cm})^2 : (0.4 \text{ km})^2 = 1 \text{ cm}^2 : 0.16 \text{ km}^2$$

Step 3. (a) Map → ground, multiply by $0.16$:

$$125 \text{ cm}^2 \times 0.16 = 20 \text{ km}^2$$

Step 4. (b) Ground → map, divide by $0.16$:

$$\dfrac{4.8}{0.16} = 30 \text{ cm}^2$$

Check. Consistency between the parts: $30$ cm² should be less than $125$ cm² because $4.8$ km² is less than $20$ km². ✓

⚠ Watch out: Areas scale by the square of the length scale — using $0.4$ instead of $0.16$ is the error every map-area question is built to catch. Convert the scale to sensible units before squaring ($0.4$ km is easier to square than $40\,000$ cm), and square the units along with the number.