A representative fraction (RF) such as 1:40 000 tells you how much the map has been shrunk. To find a ground distance, multiply the map distance by the second number, then convert the units.
This lesson is part of topographic scale, distance and area. For the wider picture of scale, direction and grid references, see scale, direction and grid references.
What does a representative fraction tell you?
An RF compares one unit on the map with the same unit on the ground. On a map of 1:40 000, 1 cm on paper is 40 000 cm on the land.
Forty thousand centimetres is 400 m, so the same scale can be written as a statement: “1 cm represents 400 m”. Statement scales are easier to use for quick checks.
Three steps from RF to kilometres
- Measure the map distance in centimetres.
- Multiply by the second number of the RF. The answer is in centimetres on the ground.
- Divide by 100 to get metres, then by 1 000 to get kilometres.
Write the unit after each number. The unit tells you which conversion is still missing.
Worked example: an original bridge and jetty
An original map has an RF of 1:40 000. A bridge and a jetty are 5.5 cm apart on the map. Find the ground distance in kilometres.
Step 2: 5.5 × 40 000 = 220 000 cm.
Step 3: 220 000 ÷ 100 = 2 200 m, and 2 200 ÷ 1 000 = 2.2 km.
A quick statement check agrees. At 1 cm to 400 m, 5.5 cm is 5.5 × 400 = 2 200 m.
Working backwards from the ground
Some questions give the ground distance and ask for the map distance. Reverse the steps: change the ground distance to centimetres, then divide by the second number.
On a map of 1:50 000, a trail is 3 km long. That is 3 × 1 000 × 100 = 300 000 cm. Divide by 50 000 to get 6 cm on the map.
The mistake that shifts the answer
The common slip is to divide 220 000 cm by 1 000 and write 220 km. The arithmetic is tidy, but it skips the step from centimetres to metres.
| Line | Wrong | Right |
|---|---|---|
| Ground distance | 220 000 cm | 220 000 cm |
| Convert | ÷ 1 000 | ÷ 100 000 |
| Answer | 220 km | 2.2 km |
A 220 km walk between a bridge and a jetty on one sheet of map is impossible. A sense check on size catches most slips of this kind.
Check yourself
An original map has an RF of 1:25 000. A school and a clinic are 8.4 cm apart. Find the ground distance in kilometres and write the scale as a statement.
Answer
8.4 × 25 000 = 210 000 cm.
210 000 ÷ 100 = 2 100 m, so the distance is 2.1 km.
The statement scale is 25 000 cm = 250 m, so “1 cm represents 250 m”. Check: 8.4 × 250 = 2 100 m.
What to study next
Next, see how a bending road changes the method in measuring a winding route without treating it as a straight line. Then test yourself with the scale, distance and area practice set, and try your own numbers in the map scale distance and area checker.
If you want a teacher to watch your conversions line by line, see online one-to-one Geography tuition.