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11.05 Scales on maps and diagrams

Lesson

So far we have used ratios in a wide range of contexts, but they are particularly useful in scale drawings. To create maps, building plans, and other technical drawings, the features being represented must be scaled down to fit on the piece of paper, and we express this scaling factor with a ratio. For example, if a small city is $100000$100000 times larger than a piece of paper, scaling its features down onto a map drawn on that paper would have the scaling ratio of $1:100000$1:100000, meaning $1$1 cm measured on the map represents $100000$100000 cm (or $1$1 km) in real life. 

For a scale ratio in the form $1:A$1:A, $A$A is known as the scale factor. For example, a drawing with the scale $1:20$1:20, has a scale factor of $20$20 and indicates that lengths in the real life object will be $20$20 times larger than lengths in the drawing.

Another way to represent the distances on a map or building plan is to use a scale bar. This small bar on the drawing shows the corresponding distance in real life. On a map, a scale bar might measure $10$10 cm long, but if it is labelled as $20$20 km we know that if two features are $10$10 cm apart on the map then they are $20$20 km apart in real life.

Scale diagrams

$\text{actual length}=\text{drawn length}\times\text{scale factor}$actual length=drawn length×scale factor

$\text{drawn length}=\text{actual length}\div\text{scale factor}$drawn length=actual length÷scale factor

 

Worked examples

Example 1

Represent the following description as a scale ratio: $5$5 cm on the map represents $25$25 m in real life.

Think: We need to have our two quantities in the same unit of measurement for this ratio. Let's convert the $25$25 m to centimetres.

Do:

$25$25 m $=$= $25\times100$25×100 cm
  $=$= $2500$2500 cm

Once we have equivalent quantities, we can write the scale as a ratio and simplify if possible.

$5:2500$5:2500 $=$= $1:500$1:500

Hence, the scale ratio for this map is $1:500$1:500, indicating a scale factor of $500$500.

Example 2

The following is a scale diagram of a room using the scale $1:50$1:50.

(a) Find the actual length and width of the room life in metres.

Think: The scale factor of the diagram is $50$50, so the real lengths will be $50$50 times larger. Let's find the lengths and then convert to metres.

Do:

Actual length $=$= $19\times50$19×50 cm  
  $=$= $950$950 cm

Divide by $100$100 to convert to metres.

  $=$= $9.5$9.5 m  
       
Actual width $=$= $11.5\times50$11.5×50 cm  
  $=$= $575$575 cm

Divide by $100$100 to convert to metres.

  $=$= $5.75$5.75 m  

(b) Find the actual area of the room in square metres.

Think: We have just calculated the actual length and width of the room in metres, so we can use these to find the area of the room.

Do:

Area $=$= $L\times W$L×W
  $=$= $9.5\times5.75$9.5×5.75 m2
  $=$= $54.625$54.625 m2

Caution: Notice we have used the converted lengths to find the area. However, we could have used the lengths in the diagram to find the area and then scale the area. To scale the area after the calculation be careful to multiply by the $\left(\text{Scale factor}\right)^2$(Scale factor)2, since both length and width must be scaled. Just as when we converted units of area we multiplied by the $\left(\text{Conversion factor for lengths}\right)^2$(Conversion factor for lengths)2.

(c) What percentage of the room is taken up by the blue rug?

Think: We can calculate the percentage area the blue rug takes up in the scale diagram. The fraction covered by the rug will not change if we scale the picture up or down. 

Do: Find the fraction of the area of the rug over the area of the room and multiply by $100%$100% to express as a percentage.

Percentage rug of room $=$= $\frac{\text{area rug}}{\text{area room}}\times100%$area rugarea room×100%
  $=$= $\frac{4\times2.5}{19\times11.5}\times100%$4×2.519×11.5×100%
  $=$= $\frac{10}{218.5}\times100%$10218.5×100%
  $\approx$ $4.58%$4.58%

(d) If a window that is actually $2$2 m wide was shown on the diagram, how many centimetres wide would the window be in the diagram?

Think: This time we have the actual length and want the length in the drawing, this will be $50$50 times smaller.

Do:

Drawn length $=$= $\text{Actual length}\div\text{Scale factor}$Actual length÷Scale factor  
  $=$= $2\div50$2÷50 m  
  $=$= $0.04$0.04 m

Multiply by $100$100 to convert to metres.

  $=$= $4$4 cm  

 

Practice questions

Question 1

The scale on a map of a garden is $1:2000$1:2000. How far apart on the map should two fountains be drawn if the actual distance between the fountains is $100$100 metres?

Give your answer in centimetres.

Question 2

Bianca is looking over a map of her local area and notices that the scale of the map is given as $1:100$1:100 in the map legend.

  1. Find the actual distance (in centimetres) between two points which are drawn $12$12 cm apart on that map.

  2. Find the distance in metres between the two points from part(a).

Question 3

Convert the following description to a proper scale ratio:

$3$3 cm represents an actual distance of $9$9 m.

Question 4

A map of a town is drawn to scale below.

  1. What is the distance between house $A$A and the park?

  2. What is the distance between house $C$C and the park?

  3. Find the length of the shorter side of the park.

  4. Find the length of the longer side of the park.

  5. What is the distance between house $A$A and house $C$C, in metres, by travelling along the road?

  6. What is this distance in kilometres?

Outcomes

ACMEM070

use ratio to describe simple scales

ACMEM083

use scales to find distances, such as on maps; for example, road maps, street maps, bushwalking maps, online maps and cadastral maps

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