Proportional Reasoning-2 · Lesson 2 of 7
Ratios in Maps
“Learn how map scales connect distances on a map with real geographical distances and how to convert between them correctly.”
- Explain what a representative fraction on a map means.
- Convert a map distance into an actual geographical distance.
- Convert an actual distance into the corresponding map distance.
- Use consistent units when working with map scales.
- Distinguish straight-line geographical distance from road distance.
Reading a Map Scale as a Ratio
A map is much smaller than the real region it represents, so every distance has been reduced by a fixed scale factor. A map scale written as a ratio tells us exactly how a measured distance on the page corresponds to a real distance on the ground.
A ratio that compares a distance on a map with the corresponding actual geographical distance, using the same unit for both quantities.
An RF of 1 : 60,00,000 means that 1 unit on the map represents 60,00,000 of the same units on the ground. If the map distance is measured in centimetres, then 1 cm on the map represents 60,00,000 cm in reality.
Problem
A map has RF 1 : 60,00,000. What actual distance does 1 cm on the map represent in kilometres?
- 1.The scale says 1 cm on the map represents 60,00,000 cm on the ground.
- 2.Since 100 cm = 1 m, divide by 100: 60,00,000 cm = 60,000 m.
- 3.Since 1000 m = 1 km, divide by 1000: 60,000 m = 60 km.
- 4.So 1 cm on the map represents 60 km geographically.
Finding Actual Distance
Once the scale has been interpreted, finding the actual distance is a proportional calculation. Multiply the measured map distance by the scale factor, then convert the resulting length into a convenient unit such as kilometres.
Problem
Two locations are 3.2 cm apart on a map with RF 1 : 60,00,000. What is their approximate geographical distance?
- 1.Each 1 cm represents 60 km, as shown by the scale conversion.
- 2.For 3.2 cm, multiply 3.2 × 60 = 192.
- 3.The approximate geographical distance is 192 km.
A ruler measures the straight-line separation between points on the map. A road may bend around rivers, hills or settlements, so the road distance can be longer than the geographical distance obtained from the map scale.
Finding Map Distance
We can also work backwards. If we know the real distance and the scale, we divide by the distance represented by 1 cm to find how far apart the points should appear on the map.
Problem
On a map, 1 cm represents 5 km. How far apart should two places that are 37.5 km apart be drawn?
- 1.Write the relationship: 1 cm represents 5 km.
- 2.Divide the actual distance by 5: 37.5 ÷ 5 = 7.5.
- 3.The two points should be 7.5 cm apart on the map.
Making a Scale Drawing
The same idea can be used to draw a classroom, room or playground accurately on paper. With a scale of 1 : 50, every 50 cm in the real room becomes 1 cm in the drawing. Objects keep their relative positions and sizes only when the same scale is used throughout.
Problem
A classroom wall is 8 m long. How long should it be drawn at a scale of 1 : 50?
- 1.Convert 8 m to 800 cm so both distances use centimetres.
- 2.At 1 : 50, divide the real length by 50: 800 ÷ 50 = 16.
- 3.Draw the wall 16 cm long.
Do not compare 1 cm directly with a number written in kilometres inside an RF calculation. The RF is unitless only because both sides are understood to use the same unit. Convert units carefully before calculating.
Quiz
What does RF 1 : 2,00,000 mean?
If 1 cm on a map represents 60 km, what does 2.5 cm represent?
Why can a road distance be longer than the distance calculated from a map ruler?
At scale 1 : 50, a 3 m table should be how long in the drawing?
A map uses 1 cm : 8 km. A real distance of 28 km should appear as how many centimetres?
Practice Problems
- A map has RF 1 : 50,00,000. Convert the distance represented by 1 cm into kilometres.
- Two towns are 4.6 cm apart on a map where 1 cm represents 25 km. Find their approximate geographical distance.
- Two points are 84 km apart. On a map where 1 cm represents 12 km, how far apart should they be drawn?
- A classroom is 7.5 m by 6 m. Draw a plan at scale 1 : 50 and state the dimensions of the rectangle on paper.
- Measure the same pair of locations on two maps with different scales. Explain why the calculated geographical distances should be close but may not be exactly equal.
- Explain in your own words why a map-scale calculation usually gives geographical distance rather than the length of the road route.
Key Takeaways
A map scale compares map distance with actual geographical distance. RF 1 : n means 1 unit on the map represents n of the same units on the ground. Convert units carefully before or after using the scale. Actual distance is found by multiplying map distance by the scale factor. Map distance is found by dividing actual distance by the distance represented by one map unit. A straight-line map distance is not necessarily the same as road distance.