Solved Exercises on Understanding Scale Drawings in Grade 7

Master understanding scale drawings: scale factors, actual dimensions, map scales, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Scale Factor
Exercise 1
A rectangle is drawn to scale with dimensions 3 cm by 4 cm. If the scale factor is 1:5, what are the actual dimensions?
Definition:

Scale factor: The ratio of a dimension in a scale drawing to the corresponding actual dimension. Scale 1:5 means 1 unit on the drawing represents 5 units in reality.

Scale factor method:
  1. Identify the scale factor (drawing : actual)
  2. Multiply each drawing dimension by the scale factor
  3. Calculate the actual dimensions
  4. Verify the relationship
Given Information
Drawing: 3 cm × 4 cm, Scale: 1:5
Length Calculation
3 × 5 = 15 cm
Width Calculation
4 × 5 = 20 cm
Step 1: Understand the scale factor

Scale 1:5 means every 1 cm on the drawing represents 5 cm in reality

Step 2: Calculate actual length

Drawing length = 3 cm

Actual length = 3 × 5 = 15 cm

Step 3: Calculate actual width

Drawing width = 4 cm

Actual width = 4 × 5 = 20 cm

Step 4: Verify the scale

Scale check: 3:15 = 1:5 ✓

Scale check: 4:20 = 1:5 ✓

Actual dimensions: 15 cm × 20 cm
Final answer:

The actual dimensions are 15 cm by 20 cm.

Applied rules:

• Scale multiplication: Actual = Drawing × Scale factor

• Proportional reasoning: Dimensions maintain the same ratio

• Unit consistency: Maintain same units throughout calculation

2 Map Scale Problems
Exercise 2
On a map with a scale of 1 inch = 25 miles, two cities are 3.2 inches apart. What is the actual distance between the cities?
Definition:

Map scale: A ratio that shows the relationship between distances on a map and actual distances in the real world.

Given Information
Scale: 1 inch = 25 miles, Map distance: 3.2 inches
Set up Proportion
1/25 = 3.2/x
Calculate Distance
x = 3.2 × 25 = 80 miles
Step 1: Identify the scale relationship

1 inch on map = 25 miles in reality

Step 2: Set up the proportion

(Map distance)/(Actual distance) = (Scale map)/(Scale actual)

3.2/x = 1/25

Step 3: Cross multiply and solve

3.2 × 25 = 1 × x

80 = x

Step 4: State the answer

The actual distance is 80 miles

Step 5: Verify the solution

Check: 3.2 inches ÷ 80 miles = 1 inch ÷ 25 miles ✓

80 miles
Final answer:

The actual distance between the cities is 80 miles.

Applied rules:

• Proportional reasoning: Ratios remain constant

• Cross multiplication: Solve proportions efficiently

• Unit conversion: Maintain dimensional consistency

3 Finding Scale Factor
Exercise 3
A building is 120 feet tall in reality. Its model is 8 inches tall. What is the scale factor of the model to the actual building?
Definition:

Scale factor: The ratio of the model dimension to the actual dimension, expressed in the same units.

Given Information
Actual: 120 ft, Model: 8 in
Convert Units
120 ft = 1440 in
Calculate Scale
8:1440 = 1:180
Step 1: Convert to same units

Model: 8 inches

Actual: 120 feet = 120 × 12 = 1440 inches

Step 2: Set up the ratio

Scale factor = Model dimension : Actual dimension

Scale factor = 8 : 1440

Step 3: Simplify the ratio

Find GCD of 8 and 1440

GCD(8, 1440) = 8

8 ÷ 8 = 1

1440 ÷ 8 = 180

Step 4: Write the scale factor

Scale factor = 1 : 180

Step 5: Verify the scale

Check: 1 inch represents 180 inches

So 8 inches represents 8 × 180 = 1440 inches = 120 feet ✓

Scale factor: 1:180
Final answer:

The scale factor of the model to the actual building is 1:180.

Applied rules:

• Unit conversion: Always use same units when comparing

• Ratio simplification: Reduce to lowest terms

• Scale notation: Model : Actual format

Scale Drawings Rules and Methods
\text{Scale factor} = \frac{\text{Drawing dimension}}{\text{Actual dimension}}
Scale Factor Formula
Scale Factor
Drawing : Actual
Ratio of drawing to actual size
Proportional Relationship
\(\frac{\text{Drawing}}{\text{Actual}} = \frac{\text{Drawing}}{\text{Actual}}\)
Corresponding dimensions are proportional
Unit Conversion
1 ft = 12 in, 1 mi = 5280 ft
Convert to same units before calculating
Key definitions:

Scale drawing: A drawing that represents an object with accurate proportions but different size than the actual object.

Scale factor: The ratio that compares the size of the drawing to the size of the actual object.

Scale ratio: A comparison written in the form 1:x or x:1 showing the relationship between drawing and actual sizes.

Proportional: Having the same ratio between corresponding parts of different sizes.

Enlargement: A scale factor greater than 1, making the drawing larger than the actual object.

Reduction: A scale factor less than 1, making the drawing smaller than the actual object.

Scale calculation methodology:
  1. Identify given information: Determine what's known (scale, drawing size, or actual size)
  2. Convert units if needed: Ensure all measurements use the same units
  3. Set up proportion: Create ratio relationship between known and unknown
  4. Solve for unknown: Use cross multiplication or direct calculation
  5. Verify answer: Check that the scale relationship is maintained
Tip 1: Always convert measurements to the same units before calculating scale factors.
Tip 2: Remember that scale factors are ratios - they compare sizes, not add or subtract.
Tip 3: To find actual size: multiply drawing size by the scale factor.
Tip 4: To find drawing size: divide actual size by the scale factor.
Common errors: Forgetting to convert units, mixing up drawing and actual dimensions in ratios, misinterpreting scale notation, not maintaining proportional relationships, arithmetic errors in cross multiplication.
Applications: Maps and navigation, architectural blueprints, engineering drawings, model making, photography, cartography, urban planning, and any field requiring scaled representations.
Essential rules:

• Scale factor = Drawing dimension ÷ Actual dimension

• Proportional relationship: Drawing₁/Actual₁ = Drawing₂/Actual₂

• Same units required for all measurements

• Scale 1:5 means drawing is 1/5th of actual size

• Scale 2:1 means drawing is 2 times actual size

Solution: Exercises 4 to 5
4 Area Scale Problems
Exercise 4
A rectangular room measures 6 cm by 8 cm in a scale drawing with a scale of 1 cm = 2 m. What is the actual area of the room?
Definition:

Area scaling: When scaling areas, the scale factor is squared because area involves two dimensions.

Given Information
Drawing: 6 cm × 8 cm, Scale: 1 cm = 2 m
Convert Dimensions
12 m × 16 m
Calculate Area
192 m²
Step 1: Convert drawing dimensions to actual dimensions

Length: 6 cm × 2 m/cm = 12 m

Width: 8 cm × 2 m/cm = 16 m

Step 2: Calculate actual area

Area = Length × Width

Area = 12 m × 16 m = 192 m²

Step 3: Alternative method using area scale

Area scale factor = (Linear scale factor)²

Area scale factor = 2² = 4

Drawing area = 6 × 8 = 48 cm²

Actual area = 48 × 4 = 192 m²

Step 4: Verify the solution

Check: 12 m × 16 m = 192 m² ✓

192 m²
Final answer:

The actual area of the room is 192 square meters.

Applied rules:

• Area scaling: Area scale factor = (Linear scale factor)²

• Dimension conversion: Convert linear dimensions first

• Two methods: Either convert dimensions then calculate area, or use area scale factor

5 Real-World Application
Exercise 5
An architect is designing a house. The blueprint uses a scale of 1 inch = 4 feet. If the living room is to be 20 feet by 24 feet, what should the dimensions be on the blueprint?
Definition:

Real-world applications: Scale drawings are essential in architecture, engineering, and design for planning and visualization.

Given Information
Scale: 1 in = 4 ft, Actual: 20 ft × 24 ft
Calculate Length
20 ÷ 4 = 5 in
Calculate Width
24 ÷ 4 = 6 in
Step 1: Understand the scale relationship

1 inch on blueprint = 4 feet in reality

This means: 1 foot in reality = 1/4 inch on blueprint

Step 2: Calculate blueprint length

Actual length = 20 feet

Blueprint length = 20 feet × (1/4 inch per foot) = 5 inches

Step 3: Calculate blueprint width

Actual width = 24 feet

Blueprint width = 24 feet × (1/4 inch per foot) = 6 inches

Step 4: Verify the scale

Check: 5 in : 20 ft = 1 in : 4 ft ✓

Check: 6 in : 24 ft = 1 in : 4 ft ✓

Step 5: State the answer

The blueprint dimensions should be 5 inches by 6 inches

Blueprint dimensions: 5 in × 6 in
Final answer:

The dimensions on the blueprint should be 5 inches by 6 inches.

Applied rules:

• Reverse scaling: When going from actual to drawing, divide by scale factor

• Proportional reasoning: Maintain the same ratio in all dimensions

• Real-world context: Understand the practical application of scale drawings

Detailed Scale Drawings Guide
\frac{\text{Drawing dimension}}{\text{Actual dimension}} = \text{Scale factor}
Scale Ratio Formula
Key definitions:

Scale drawing: A representation of an object where all dimensions are proportionally reduced or enlarged compared to the actual object.

Scale factor: The constant of proportionality that relates drawing dimensions to actual dimensions.

Proportional relationship: A relationship where the ratio of corresponding dimensions remains constant.

Similar figures: Figures that have the same shape but different sizes, with corresponding angles equal and corresponding sides proportional.

Linear scale: The ratio of any linear measurement in the drawing to the corresponding actual measurement.

Area scale: The square of the linear scale factor, used when comparing areas.

Complete scale drawing methodology:
  1. Identify the scale: Determine the relationship between drawing and actual measurements
  2. Convert units if necessary: Ensure all measurements use consistent units
  3. Set up proportional relationship: Create equivalent ratios between drawing and actual dimensions
  4. Solve for unknown: Use cross multiplication or direct calculation
  5. Verify solution: Check that the scale relationship holds for all corresponding dimensions
Tip 1: Always write the scale factor as Drawing:Actual or Drawing/Actual to avoid confusion.
Tip 2: When scaling areas, remember to square the linear scale factor.
Tip 3: Draw a small sketch to visualize the problem and identify corresponding dimensions.
Tip 4: Check your answer by ensuring the scale ratio is maintained for all dimensions.
Common errors: Mixing up drawing and actual dimensions, forgetting to convert units, not squaring scale factors for area calculations, misinterpreting scale notation, not maintaining proportional relationships.
Applications: Architecture and construction, engineering design, map-making and navigation, model building, photography and art, surveying, and any field requiring scaled representations of real objects.
Essential scale rules:

• Scale factor = Drawing dimension ÷ Actual dimension

• Corresponding dimensions are proportional

• Area scale factor = (Linear scale factor)²

• Volume scale factor = (Linear scale factor)³

• Scale 1:10 means drawing is 1/10th of actual size

• Scale 5:1 means drawing is 5 times actual size

Scale Drawings Guide

📊
Scale Factor

1:5 = Drawing is 1/5 of actual

2:1 = Drawing is 2 times actual

Ratio of drawing to actual size

Proportions

Drawing/Actual = Scale

Corresponding dimensions proportional

Same shape, different size

Area Scaling

Area scale = (Linear scale)²

2D measurements squared

Scale factor applied twice

Verification

Check all dimensions

Verify same scale ratio

Convert units consistently

Questions & Answers

Question: How do I know if I should multiply or divide when working with scale factors?

Answer: The key is to think about the relationship and what you're trying to find:

Going from drawing to actual (making bigger):

  • If you have the drawing dimension and want the actual dimension
  • Use: Actual = Drawing × Scale factor
  • Example: Scale 1:10, Drawing = 5 cm → Actual = 5 × 10 = 50 cm

Going from actual to drawing (making smaller):

  • If you have the actual dimension and want the drawing dimension
  • Use: Drawing = Actual ÷ Scale factor
  • Example: Scale 1:10, Actual = 50 cm → Drawing = 50 ÷ 10 = 5 cm

Memory trick:

  • Scale 1:10 means drawing is 1/10th of actual
  • To get from small (drawing) to big (actual), multiply
  • To get from big (actual) to small (drawing), divide

Always consider: Am I going from small to big or big to small?

Question: Why do I need to square the scale factor when working with areas?

Answer: This happens because area involves TWO dimensions, so the scale factor affects both length and width!

Example: Rectangle with scale 1:3

  • Drawing: 2 cm × 4 cm → Area = 8 cm²
  • Actual: (2×3) cm × (4×3) cm = 6 cm × 12 cm → Area = 72 cm²

Comparison:

  • Linear scale: 1:3
  • Area comparison: 8:72 = 1:9 = 1:3²

Why this happens:

  • Length is scaled by the scale factor
  • Width is also scaled by the scale factor
  • Area = Length × Width, so both dimensions are affected
  • Area scale factor = (Linear scale factor) × (Linear scale factor) = (Linear scale factor)²

For volumes, it would be cubed since volume involves three dimensions!

Question: What if the scale is given as a fraction like 1/4 inch = 1 foot? How do I work with that?

Answer: Fractional scales work the same way! Just convert to a consistent form:

Scale: 1/4 inch = 1 foot

Method 1 - Convert to integer ratio:

  • Multiply both sides by 4: 1 inch = 4 feet
  • Now you have 1:4 (but with different units)
  • This means 1 inch on drawing = 4 feet in reality

Method 2 - Use as is:

  • Set up proportion: (1/4 inch)/(1 foot) = (drawing)/(actual)
  • For example: If actual = 12 feet, then drawing = (1/4) × (12/1) = 3 inches

Converting fractional scales:

  • 1/2 inch = 1 mile → 1 inch = 2 miles
  • 1/8 inch = 1 inch → 1 inch = 8 inches (or 1:8 ratio)
  • Always aim for the same unit on the drawing side for consistency

The key is maintaining the proportional relationship!