3.07 Proportions with maps and scale
Introduction
Ratios and proportions are everywhere. You might have seen a scale on a map. Scale drawings make it easy to see large figures such as buildings and roads, on paper. We will now apply the concept of ratio and proportion with maps and scales.
Proportions with maps and scales
An architectural floor plan is an example of a scale drawing. A scale drawing is a diagram of an object in which the dimensions are in proportion to the actual dimensions of the object.
The scale of a scale drawing tells us how the drawing's dimensions and the real-life dimensions are related. For example, the scale 1 \text{ in.} : 12 \text{ ft.} tells us that 1 inch in the floor plan drawing represents 12 feet in the actual building.
A graphic scale represents a scale by using a small line with markings similar to a ruler. One side of the line represents the distance on the map, while the other side represents the true distances of objects in real life. By measuring the distance between two points on a map and then referring to the graphic scale, we can calculate the actual distance between those points. In this picture, you can see that 1 cm on the scale represents 250 kilometers in real life and 1.5 cm on the scale represents 250 miles in real life.
A verbal scale uses words to describe the ratio between the map's scale and the real world. For example, we could say "One inch equals fifteen miles" or we could write it as 1 \text{ in.} = 15 \text{ miles}. This means that one inch on the map is equivalent to 15 miles in the real world.
The examples above include scales with units. A scale with two dimensions listed in the same unit, such as 1 \text{ in.}:48 \text{ in.}, can be written without units, such as 1:48. This is called the scale ratio.
When a ratio is fully simplified, we call it a simplified scale ratio.
| Scale with units | Scale without units |
|---|---|
| 1 \text{ in.} : 4 \text{ ft.} | 1 : 48 |
The ratio of 1 : 48 can also be written as the fraction \dfrac{1}{48}. This fraction is called the scale factor. It tells us that the dimensions of the model are \dfrac{1}{48}\text{th} the size of the dimensions in real life.
Here are some helpful unit conversions before we get started with examples.
\begin{aligned} 1 \text{ ft} &= 12 \text{ in} \\ 1 \text{ yd} &= 3 \text{ ft or } 36 \text{ in} \\ 1 \text{ mi} &= 1 \, 760 \text{ yd} \end{aligned}
\begin{aligned} 1 \text{ cm} &= 10 \text{ mm} \\ 1 \text{ m} &= 100 \text{ cm} \\ 1 \text{ km} &= 1 \, 000 \text{ m} \end{aligned}
Examples
Example 1
Convert the following description to a simplified scale ratio:
3 cm represents an actual distance of 9 m.
Example 2
Bianca is looking over a map of her local area and notices that the scale of the map is given as
1 : 100 in the map legend.
Find the actual distance (in centimeters) between two points which are drawn 12 cm apart on that map.
Find the distance in meters between the two points from part (a).
Example 3
The scale ratio shown on a map is 1 : 500\,000. How far apart on the map should two train stations be drawn if the actual distance between the stations is 100 miles?
Using the fact 1 \text{ mile} = 63\,360 \text{ inches}, express your answer in inches, rounding to two decimal places.
A scale drawing is a diagram of an object in which the dimensions are in proportion to the actual dimensions of the object.
The scale of a scale drawing tells us how the drawing's dimensions and the real-life dimensions are related.
A scale with two dimensions listed in the same unit and can be written without units is called the scale ratio.
When a ratio is fully simplified, we call it a simplified scale ratio.