Reading a Map Scale

A map has a scale of $1:50,000$. Two cities are 4 cm apart on the map. What is the actual distance?

Understand the scale: $1:50,000$ means 1 cm on map = 50,000 cm in reality = Scale factor = 50,000

Multiply drawing measurement by scale factor: $4 \text{ cm} \times 50,000 = 200,000 \text{ cm}$ = 200,000 cm

Convert to appropriate units: $200,000 \text{ cm} \div 100 = 2,000 \text{ m}$ $2,000 \text{ m} \div 1,000 = 2 \text{ km}$ = 2 km

Answer: The actual distance between the cities is 2 km.

Creating a Floor Plan

A room is 6 m by 4 m. Draw it using a scale of $1:50$. What will the dimensions be on paper?

Convert room dimensions to cm: $6 \text{ m} = 600 \text{ cm}$ $4 \text{ m} = 400 \text{ cm}$ = 600 cm by 400 cm

Divide by scale factor: $600 \div 50 = 12 \text{ cm}$ $400 \div 50 = 8 \text{ cm}$ = 12 cm by 8 cm

Check with proportion: $\frac{12}{600} = \frac{1}{50}$ ✓ $\frac{8}{400} = \frac{1}{50}$ ✓ = Proportions match

Answer: The floor plan dimensions will be 12 cm by 8 cm.

Finding the Scale Factor

A model car is 25 cm long. The actual car is 5 m long. What is the scale of the model?

Convert to same units: Actual car: $5 \text{ m} = 500 \text{ cm}$ = 500 cm

Set up the ratio: Model : Actual = $25 : 500$ = 25:500

Simplify the ratio: $\frac{25}{500} = \frac{1}{20}$ = 1:20

Write the scale: Scale = $1:20$ = Model is 1/20 of actual size

Answer: The model car is built at a scale of $1:20$.

Blueprint Measurements

On a blueprint at scale $1:75$, a wall measures 8 cm. A door opening is 1.2 cm wide. What are the actual dimensions?

Calculate actual wall length: $8 \text{ cm} \times 75 = 600 \text{ cm} = 6 \text{ m}$ = Wall: 6 m

Calculate actual door width: $1.2 \text{ cm} \times 75 = 90 \text{ cm} = 0.9 \text{ m}$ = Door: 90 cm

Verify reasonableness: A 6 m wall and 90 cm door are realistic dimensions = Makes sense!

Answer: The actual wall is 6 m long, and the door opening is 90 cm (0.9 m) wide.

Mistake: Multiplying when you should divide (or vice versa)

Why: Going from drawing to actual: multiply by scale factor. Going from actual to drawing: divide by scale factor.

Correct: Remember: Drawings are SMALLER, so drawing measurements multiplied by scale = actual size.

Mistake: Forgetting to use the same units

Why: Mixing cm and m in calculations leads to answers that are off by factors of 100.

Correct: Always convert to the SAME units before calculating, then convert the answer to appropriate units.

Mistake: Confusing $1:50$ with $50:1$

Why: $1:50$ means the drawing is smaller (1/50 size). $50:1$ would mean the drawing is 50 times bigger (enlargement).

Correct: $1:n$ means reduce by factor of $n$. $n:1$ means enlarge by factor of $n$.

Navigation and Maps

Hikers, pilots, and sailors use scale maps to plan routes and calculate distances.

A hiking map at $1:25,000$ scale means 4 cm on the map = 1 km of actual hiking distance.

Interior Design

Interior designers create floor plans to help clients visualize furniture placement.

A floor plan at $1:20$ scale shows a sofa that measures 10 cm long. The actual sofa is 200 cm (2 m) long.

Model Building

Model enthusiasts build miniature versions of cars, planes, trains, and buildings.

A $1:72$ scale model airplane of a Boeing 747 (70 m wingspan) has a model wingspan of about 97 cm.

Q: What is the difference between scale and scale factor?

A: The scale is the ratio (like $1:50$). The scale factor is the number you multiply or divide by (in this case, 50).

Q: How do I know whether to multiply or divide?

A: Drawing to actual: multiply (drawings are smaller). Actual to drawing: divide (to make it smaller).

Q: Can scales be written as fractions?

A: Yes! A scale of $1:50$ can also be written as $\frac{1}{50}$. This means the drawing is $\frac{1}{50}$ of the actual size.