145. Scale Drawings and Real Distances

Learning Intentions

• To understand that scale drawings can be used to depict large or small objects
• convert from a distance on a map or diagram to the actual distance in real life
• convert from the actual distance in real life to a distance on a map or diagram
• determine the scale factor given a distance on a diagram and the distance in real life

Pre-requisite Summary

• Know that length can be measured in units such as mm, cm, m and km
• Be able to convert between metric units of length
• Understand that a ratio compares two quantities
• Be able to simplify ratios
• Know that multiplication and division can be used to scale quantities up or down
• Understand that diagrams do not always show real size
• Be able to read labelled lengths on a map or drawing

Worked Examples

Worked Example 1

State what each scale means in words:
a) $1:100$
b) $1:50000$
c) $2:1$

Worked Example 2

A map has scale $1:100000$. Find the actual distance if the map distance is:
a) $3$ cm
b) $7.5$ cm

Worked Example 3

A floor plan has scale $1:50$. Find the actual distance if the plan distance is:
a) $4$ cm
b) $12.6$ cm

Worked Example 4

A model car is built to scale $1:20$. Find the model length if the real car length is:
a) $4$ m
b) $3.6$ m

Worked Example 5

A map distance of $5$ cm represents $2$ km in real life. Find the scale factor.

Worked Example 6

A diagram distance of $8$ cm represents a real length of $6$ m. Find the scale factor.

Problems

Problem 1

State what each scale means in words:
a) $1:200$
b) $1:25000$
c) $5:1$

Problem 2

A map has scale $1:200000$. Find the actual distance if the map distance is:
a) $2$ cm
b) $6.5$ cm

Problem 3

A floor plan has scale $1:100$. Find the actual distance if the plan distance is:
a) $5$ cm
b) $9.4$ cm

Problem 4

A model plane is built to scale $1:40$. Find the model length if the real plane length is:
a) $8$ m
b) $12$ m

Problem 5

A map distance of $4$ cm represents $3$ km in real life. Find the scale factor.

Problem 6

A diagram distance of $6$ cm represents a real length of $9$ m. Find the scale factor.

Exercises

Understanding and Fluency

1. Complete each statement:
   a) A scale drawing shows an object at a different ______
   b) In the scale $1:100$, $1$ unit on the drawing represents _ units in real life
   c) To use a scale correctly, the units must first be made the ______

2. State what each scale means in words:
   a) $1:10$
   b) $1:500$
   c) $1:100000$
   d) $3:1$

3. Convert from diagram distance to actual distance:
   a) scale $1:100$, diagram distance $6$ cm
   b) scale $1:200$, diagram distance $9$ cm
   c) scale $1:50000$, map distance $4$ cm

4. Convert from diagram distance to actual distance:
   a) scale $1:1000$, map distance $7$ cm
   b) scale $1:25$, diagram distance $12$ cm
   c) scale $1:250000$, map distance $3.2$ cm

5. Convert from actual distance to diagram distance:
   a) scale $1:100$, real length $8$ m
   b) scale $1:50$, real length $3$ m
   c) scale $1:200000$, real distance $10$ km

6. Convert from actual distance to diagram distance:
   a) scale $1:20$, real length $1.4$ m
   b) scale $1:500$, real length $25$ m
   c) scale $1:100000$, real distance $6$ km

7. Determine the scale factor in each case:
   a) $2$ cm on a map represents $1$ km in real life
   b) $5$ cm on a plan represents $10$ m in real life
   c) $4$ cm on a model represents $80$ cm in real life

8. Determine the scale factor in each case:
   a) $3$ cm on a diagram represents $12$ m in real life
   b) $8$ cm on a map represents $4$ km in real life
   c) $6$ cm on a model represents $1.2$ m in real life

Reasoning

9. Explain why units must be converted before finding a scale factor.

10. A student says that a scale of $1:100$ means you add $100$ to the drawing length to get the real length. Explain the mistake.

11. Noah says that a scale factor of $2:1$ makes the drawing smaller than the real object. Is he correct? Explain.

12. Explain why a map scale such as $1:100000$ is useful for showing large places.

13. A student finds that $5$ cm on a map represents $2$ km in real life and writes the scale as $5:2$. Describe the error.

Problem-solving

14. A map has scale $1:100000$. Two towns are $8$ cm apart on the map. Find the actual distance in kilometres.

15. A classroom plan has scale $1:50$. A wall is $14$ cm long on the plan. Find the real wall length.

16. A real road is $3.5$ km long. On a map with scale $1:50000$, how long should the road be on the map?

17. A model train is built to scale $1:25$. The real train carriage is $20$ m long. Find the model length in cm.

18. On a diagram, $6$ cm represents a real fence length of $9$ m. Determine the scale factor.

19. A map shows two landmarks $12$ cm apart. In real life they are $6$ km apart. Find the map scale.

Potential Misunderstandings

• Students may think a scale is found by comparing numbers without making the units the same
• Students may reverse the order of drawing distance and real distance in a scale ratio
• Students may multiply when they should divide, or divide when they should multiply
• Students may think a larger number in the scale always means a larger drawing
• Students may confuse enlargement scales such as $2:1$ with reduction scales such as $1:2$
• Students may forget to convert between cm, m and km when solving scale problems
• Students may write a scale factor that is not simplified
• Students may think map distances are already actual distances