Ratios and Proportions: Scale and Maps

Introduction

In this lesson, we will explore how ratios and proportions are used in scale and maps. A scale is a ratio that compares a distance on a map to the actual distance it represents.

Key Concepts

• Scale: A scale can be represented as a ratio, such as 1:100, meaning 1 unit on the map equals 100 units in reality.
• Proportions: When two ratios are equal, they form a proportion. For example, if the scale is 1:100 and you know a distance on the map, you can find the actual distance using proportions.

Example Problem

Given:

• A map shows a scale of 1:50,000.
• A distance measured on the map is 3 cm.

Steps to find the actual distance:

1. Identify the scale ratio: The scale tells us that 1 cm on the map equals 50,000 cm in reality.
2. Set up the proportion: $$\frac{1 \text{ cm}}{50,000 \text{ cm}} = \frac{3 \text{ cm}}{x}$$ where $x$ is the actual distance.
3. Cross-multiply: $$1 \times x = 3 \times 50,000$$ This simplifies to: $$x = 150,000 \text{ cm}$$
4. Convert to kilometers (if needed): Since 1 km = 100,000 cm, we convert: $$x = \frac{150,000 \text{ cm}}{100,000} = 1.5 \text{ km}$$

Conclusion

The actual distance represented by 3 cm on the map is 1.5 km.

Key Questions

• What does a scale of 1:20,000 mean in terms of actual distance?
• How can you determine the actual distance if given a different scale?
• Why is it important to understand scale when using maps?

Keywords

• Scale
• Ratio
• Proportion
• Distance
• Map