5.04 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 100\,000 times larger than a piece of paper, scaling its features down onto a map drawn on that paper would have the scaling ratio of 1:100\,000, meaning 1\text{ cm}measured on the map represents 100\,000\text{ cm} (or 1\text{ 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\text{ cm} long, but if it is labelled as 20\text{ km} we know that if two features are 10\text{ cm} apart on the map then they are 20\text{ km} apart in real life.

A scale of 1\text{:}100 will mean the the objects on the scale drawing will be 100 times smaller. That is, the true distance will be 100 times larger than the scale distance.

Examples

Example 1

The following is a 1\text{:}200 floor plan of a house. The homeowner wishes to add a dining room table, which is 150\text{ cm} long, placing it where the \times is marked on the floor plan.

Find the table's length that should be drawn to in the floor plans.

Worked Solution

Create a strategy

Use the unitary method.

Apply the idea

We want the number on the right side of the ratio to be 150. So we can divide the ratio by 200 then multiply by 150.

\displaystyle 1:200	\displaystyle =	\displaystyle \dfrac{1}{200} : \dfrac{200}{200}	Divide by 200
	\displaystyle =	\displaystyle 0.005:1	Evaluate
	\displaystyle =	\displaystyle 0.005\times 150:1\times 150	Multiply by 150
	\displaystyle =	\displaystyle 0.75 \text{:} 150	Evaluate

The first part of this ratio tells us the length of the table on the plan, in centimetres. So the length of the table that should be drawn in the floor plan is 0.75 cm.

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Example 2

The following is a 1:66\,000 scale drawing of the sailing route from the mainland to an island off the coast.

The captain approximates the distance to be 10.3\text{ cm} on the map. What is the distance of the boat trip in kilometres?

Worked Solution

Create a strategy

Use the unitary method.

Apply the idea

We want the left side of the ratio to be 10.3 so we multiply the ratio by 10.3.

\displaystyle 1:66\,000	\displaystyle =	\displaystyle 1\times 10.3:66\,000 \times 10.3	Multiply the ratio by 10.3
	\displaystyle =	\displaystyle 10.3:679\,800	Evaluate

The distance of the boat trip is 679\,800\text{ cm.} To convert this to kilometres we use the fact that 1 \text{ km} = 100\,000 \text{ cm.}

\displaystyle \text{Distance}	\displaystyle =	\displaystyle \dfrac{679\,800}{100\,000}\text{ km}	Divide by 100\,000
	\displaystyle =	\displaystyle 6.798\text{ km}	Evaluate

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Example 3

The map designer for a new amusement park measures the main street to be 4 cm. The walk along the main street is known to be 120\text{ m}.

What ratio is the map using?

Give your final answer in the form, 1: ⬚.

Worked Solution

Create a strategy

Simplify the ratio by converting to the same units and dividing by the HCF.

Apply the idea

\displaystyle \text{Ratio}	\displaystyle =	\displaystyle 4\text{ cm}:120\text{ m}	Write the ratio
	\displaystyle =	\displaystyle 4:12\,000	Convert to centimetres
	\displaystyle =	\displaystyle \dfrac{4}{4} : \dfrac{12\,000}{4}	Divide by 4
	\displaystyle =	\displaystyle 1 : 3000	Evaluate

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