Scale

Understanding the Ratio That Relates Map Distance to Actual Distance in the Real World

CAPS Grade 10 Mathematical Literacy

In Grade 10 Mathematical Literacy, Scale is the ratio that relates the distance on a map or plan to the actual distance in the real world. It allows you to represent large areas or objects on a small piece of paper while maintaining accurate proportions.

Scale Overview

Scale is a fundamental concept in map reading and plan interpretation. It enables us to understand the relationship between measurements on a representation and the actual dimensions in the real world. Mastering scale calculations is essential for navigation, construction, and planning.

Scale Concepts

Number Scale Bar Scale Ratio Proportion Actual Distance Map Distance Unit Conversion Scale Factor

Types of Scales

Scale Classification

Number Scale vs Bar Scale

Number Scale: 1 : 50,000 | Bar Scale: [=====] 0 500m 1km

There are two primary types of scales you must be able to interpret and use: Number (Ratio) Scale expressed as a ratio, and Bar (Linear) Scale shown as a visual line divided into segments.

Number (Ratio) Scale
Expressed as a ratio (e.g., 1 : 50,000). 1 unit on map represents 50,000 same units in reality.
Bar (Linear) Scale
Visual line divided into segments, each representing a specific real-world distance.
Scale Relationship
Map Distance � Scale Factor = Actual Distance. All measurements must be in same units.
Unit Conversion
100 cm = 1 m, 1,000 m = 1 km, 100,000 cm = 1 km.

Advantages and Disadvantages

Number Scale

A ratio scale expressed as numbers (e.g., 1:50,000). Easy to use for calculations.

Pros and Cons

Advantages: More accurate, easier to use, consistent ratio.
Disadvantages: Fails if map is resized, requires unit conversion.

Bar Scale

A visual line divided into segments showing real-world distances.

Pros and Cons

Advantages: Remains accurate when resized, visual and intuitive.
Disadvantages: Less precise, requires measuring bar first.

Comparison

Quick comparison of key features.

Feature Comparison

Accuracy: Number Scale wins � Resizing: Bar Scale wins � Simplicity: Number Scale wins � Visual: Bar Scale wins

Core Calculations

1

Calculate Actual Distance

Given the map distance and scale, find the real-world distance using multiplication.

Formula: Actual Distance = Map Distance � Scale Factor
Example: 5 cm � 100 = 500 cm = 5 m
2

Calculate Map Distance

Given the actual distance and scale, find the distance to draw on the map using division.

Formula: Map Distance = Actual Distance � Scale Factor
Example: 10 m = 1,000 cm, 1,000 cm � 50 = 20 cm
3

Determine the Scale

If you know both the map and actual distances, you can find the scale by dividing them.

Formula: Scale = Map Distance : Actual Distance (same units)
Example: 4 cm map, 8 m actual (800 cm) ? Scale Factor = 800 � 4 = 200 ? 1:200
4

Working with Bar Scales

When using a bar scale, first measure the bar to establish the relationship.

Method: Measure bar (e.g., 2 cm = 10 km), so 1 cm = 5 km, then multiply.
5

Unit Conversion Reference

Always ensure units are consistent before calculating.

Conversions: 10 mm = 1 cm, 100 cm = 1 m, 1,000 m = 1 km, 100,000 cm = 1 km
Common Scales: 1:100 ? 1 cm = 1 m, 1:50,000 ? 1 cm = 500 m, 1:100,000 ? 1 cm = 1 km

Scale Calculation Examples

Example 1: Finding Actual Distance

Hiking map scale 1:25,000. Distance on map is 8.4 cm.

1

8.4 � 25,000 = 210,000 cm = 2,100 m = 2.1 km

Example 2: Finding Map Distance

15 m garden path on a 1:200 plan.

1

15 m = 1,500 cm, 1,500 � 200 = 7.5 cm

Example 3: Determining Scale

Room 6 m actual, 3 cm on plan.

1

6 m = 600 cm, 600 � 3 = 200, Scale = 1:200

Example 4: Bar Scale

3 cm = 6 km on bar scale. Route measures 4.5 cm.

1

1 cm = 2 km, 4.5 � 2 = 9 km

Interactive Scale Challenge

Question 1: On a map with scale 1:100, a room measures 4.2 cm. What is the actual length in meters?
Question 2: A 12 m wall needs to be drawn on a 1:50 plan. How many cm should it be?
Question 3: A room is 5 cm on a plan but actually 10 m long. What is the scale?
Question 4: A bar scale shows 2 cm = 5 km. A route measures 3.6 cm. How long is it?
Question 5: On a 1:25,000 map, two points are 6.8 cm apart. What is the actual distance in km?

Scale Converter and Calculator

Enter values to calculate actual distances, map distances, or determine scale.

Result: 5 cm on map at scale 1:100 = 500 cm = 5 m

Scale Determination Practice

Given the map distance and actual distance, determine the scale.

Map Distance
Actual Distance
Scale (1:___ )
4 cm
8 m
2.5 cm
5 km
6 cm
3 m
8 cm
2 km

Scale Type Comparison Quiz

Choose whether each statement applies to Number Scale (N) or Bar Scale (B).

Remains accurate when map is photocopied
Expressed as a ratio (e.g., 1:50,000)
Requires measuring the scale first before use
Generally more accurate for precise calculations
A visual line divided into segments

Scale Problem-Solving Framework

I
Identify

Identify Scale Type and What's Given

Determine whether you're working with a number scale (ratio) or bar scale.

Key Questions: Is it a number scale (1:XXX) or bar scale? What are we trying to find?
C
Convert

Convert to Consistent Units

Ensure all measurements are in the same units before performing calculations.

Conversion Tips: meters ? cm (�100), km ? cm (�100,000)
S
Select

Select the Correct Formula

Choose the appropriate formula based on what you need to find.

Formulas: Actual = Map � Scale, Map = Actual � Scale, Scale = Actual � Map
C
Calculate

Perform the Calculation

Apply the formula carefully. For number scales, multiply or divide by the scale denominator.

Calculation Tips: Double-check your multiplication. Keep track of units throughout.
V
Verify

Verify and Convert to Appropriate Units

Check if your answer makes sense. Convert to practical units (meters or kilometers).

Verification: Is the answer reasonable? For a building plan, meters make sense.

Assessment Focus Areas

Number Scale Calculations

Perform accurate calculations using ratio scales.

Key Skills

  • Multiply map distance by scale factor
  • Divide actual distance by scale factor
  • Calculate scale from known distances
  • Convert units correctly

Bar Scale Interpretation

Read and use bar scales correctly.

Key Skills

  • Measure bar scale with ruler
  • Determine ratio from bar segments
  • Apply ratio to other measurements
  • Understand resizing properties

Unit Conversions

Convert between mm, cm, m, and km accurately.

Key Skills

  • Convert cm to m (�100)
  • Convert m to km (�1000)
  • Convert cm to km (�100,000)
  • Work with mixed units confidently

Choosing Scales

Understand when to use number scales versus bar scales.

Key Skills

  • Compare advantages of each scale type
  • Select appropriate scale for context
  • Explain why bar scales survive resizing
  • Recognize limitations of each type

CAPS Curriculum Requirements

Knowledge and Understanding

  • Understand the concept of scale as a ratio
  • Differentiate between number scale and bar scale
  • Know advantages and disadvantages of each scale type
  • Understand unit conversion principles

Skills and Applications

  • Calculate actual distances from map measurements
  • Determine map distances from real dimensions
  • Find the scale when given both distances
  • Interpret and use bar scales correctly

Real-World Contexts

  • Reading road maps and hiking trails
  • Working with architectural floor plans
  • Understanding engineering drawings
  • Interpreting tourist maps and guides