14.05 Maps and scale
Ratios are very useful in representing something large, like a house or a city, using a smaller drawing, called 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.
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.
A scale of $1:100$1:100 will mean the the objects on the scale drawing will be $100$100 times smaller. That is, the true distance will be $100$100 times larger than the scale distance.
Worked examples
example 1
Convert the following situation to a proper scale ratio: $5$5 cm on the map is equivalent to $25$25 m in real life.
Think: We need to have our two quantities in the same unit of measurement. Let's convert everything to centimetres. Once we have equivalent quantities, we can write the scale factor.
Do: First, convert to centimetres
| $25$25m | $=$= | $25\times100$25×100 cm |
$1$1m $=$= $100$100cm. |
| $=$= | $2500$2500 cm |
Then, rewrite the ratio of $5$5cm to $25$25 m using the the same units:
| $5$5cm : $25$25m | $=$= | $5$5cm:$2500$2500cm |
Using the conversion above. |
| $=$= | $5:2500$5:2500 |
Cancel the units. |
|
| $=$= | $1:500$1:500 |
Simplify the ratio. |
example 2
Given that the scale on a map is $1$1:$50000$50000, find the actual distance between two points that are $8$8 cm apart on the map. Give your answer in kilometres.
Think: This means that $1$1 cm on the map represents $50000$50000cm (or $500$500m) in real life.
Do: So to figure out how far $8$8cm represents, we need to multiply $8$8 by $50000$50000. Then convert to km.
| $8\times50000$8×50000 | $=$= | $400000$400000 cm |
| $=$= | $4000$4000 m | |
| $=$= | $4$4 km |
Now let's look at how we can do this process in reverse.
example 3
Given that the scale of a map of a garden is $1$1:$250$250 , how far apart should two trees be drawn on the map if they are to be planted $375$375 m apart?
Think: In this question, we are going from big units to small units, so we need to use division.
$375\div250=1.5$375÷250=1.5cm
Do: This means that the trees should be drawn $1.5$1.5 cm apart on the map.
Practice questions
question 1
The following is a $1:200$1:200 floor plan of a house.
The homeowner wishes to add a dining room table, which is $150$150 cm long, placing it where the$\times$×is marked on the floor plan.
What length should the table be drawn to in the floor plans?
Enter your answer as an unrounded decimal.
question 2
The following is a $1:66000$1:66000 scale drawing of the sailing route from the mainland to an island off the coast.
The captain approximates the distance to be $10.3$10.3 cm on the map. What is the distance of the boat trip in kilometres?
Give your answer as an unrounded decimal.
question 3
The map designer for a new amusement park measures the main street to be $4$4 cm. The walk along the main street is known to be $120$120 m.
What ratio is the map using?
Give your final answer in the form, $1:\editable{}$1:.