Lesson 6-8: Solve Problems Involving Scale Drawings

Property

A scale drawing shows an object in two dimensions with all parts in proportion to the real thing, while a scale model represents the same idea in three dimensions.
The scale is the ratio that relates the dimensions of the drawing or model to the actual dimensions of the object, written as:

Examples

• A blueprint of a house is a scale drawing. The scale might be $1 \text{ inch} : 5 \text{ feet}$.
• A toy car is a scale model of a real car. The scale might be $1:18$.
• A map is a scale drawing. The scale might be $1 \text{ cm} : 10 \text{ km}$.

Explanation

A scale drawing or model represents a real object with all its dimensions reduced or enlarged by the same factor. The scale tells you how the drawing's measurements relate to the object's actual measurements. For example, a scale of $1 \text{ inch} : 3 \text{ feet}$ means that every inch on the drawing represents three feet on the actual object. This allows for the creation of conveniently sized representations of very large or very small objects.

Property

Scale notations express the relationship between drawing measurements and actual measurements in three common formats: ratio form ($1:100$), unit form ($1 \text{ inch} = 5 \text{ feet}$), and fraction form ($\frac{1}{100}$).

Examples

Property

To find missing measurements, set up a simple fraction proportion: $\frac{\text{Drawing}}{\text{Actual}} = \frac{\text{Drawing}}{\text{Actual}}$.
Crucial Rule: If your scale does not specify units (like 1:50), you MUST make sure both your drawing and actual measurements are converted to the exact same unit before doing the math.

Examples

• Finding Actual Distance (Unit Form): A map scale is 1 cm = 10 km. Two cities are 4.5 cm apart on the map.
  • Math: 4.5 * 10 = 45. The cities are 45 km apart.
• Finding Actual Distance (Ratio Form): A model boat has a scale of 1:30. The model is 15 cm long. How long is the real boat in meters?
  • Step 1: 15 cm * 30 = 450 cm (Real boat length in cm).
  • Step 2: Convert to meters. 450 cm = 4.5 m.
• Finding Drawing Length: A room is actually 14 feet wide. The blueprint scale is 1 inch = 4 feet.
  • Math: 14 / 4 = 3.5. Draw it 3.5 inches wide on the paper.

Explanation

When solving these, always keep your labels on your numbers! If you write $\frac{1 \text{ in}}{4 \text{ ft}} = \frac{x \text{ in}}{14 \text{ ft}}$, you can clearly see that you need to cross-multiply or divide. If a problem gives you a unitless ratio like 1:300 but asks for the real answer in meters, always calculate the real answer in the tiny unit (like centimeters) first, and do the conversion at the very end.