Map and Scale Diagrams

Statement

Map and scale diagrams use ratios to represent real distances. If the scale is \(1:k\):

\[ \frac{\text{drawing}}{\text{actual}} = \frac{1}{k}, \quad \text{so} \quad \text{actual} = k \times \text{drawing} \]

Why it’s true

• A scale tells us how many times smaller (or larger) the drawing is compared to reality.
• For example, a scale of 1:100 means 1 cm on the drawing represents 100 cm in real life.
• Multiplying the drawing measurement by the scale factor gives the actual size.

Recipe (how to use it)

1. Identify the scale (e.g., 1:50, 1:1000).
2. To find the actual length, multiply the drawing length by \(k\).
3. To find the drawing length, divide the actual length by \(k\).
4. Always keep units consistent (convert cm to m, etc. if needed).

Spotting it

You’ll see this in map questions, model-making, or technical drawings where a diagram represents something larger or smaller in real life.

Common pairings

• Maps (distance conversions).
• Model building (scale replicas).
• Blueprints and engineering drawings.

Mini examples

1. Given: A map has scale 1:50,000. 3 cm on the map represents? Solution: Actual = 3 × 50,000 = 150,000 cm = 1.5 km.
2. Given: A scale drawing uses 1:20. An actual wall is 8 m long. How long on the drawing? Solution: Drawing = 800 cm ÷ 20 = 40 cm.

Pitfalls

• Wrong direction: Multiplying when you should divide (and vice versa).
• Mixing units: Always convert to the same unit first.
• Forgetting the ratio meaning: 1:100 means 1 unit, not 1 cm unless specified.

Exam strategy

• Write the formula: actual = scale × drawing.
• Convert all units before substituting.
• Check if your answer makes sense — actual should be larger than drawing if the scale factor is >1.

Summary

Scale diagrams work with the ratio \(1:k\). Formula: actual = drawing × \(k\). Divide instead to find drawing size. Always keep units consistent.