Math CentralQuandaries & Queries


Question from Dayne, a student:

My teacher doesn't now how to teach us scale factors and i have a problem to make it easier to understand. The problem is: A map of Levi's property is being made with a scale of 2 cm : 3 meters. What is the scale factor?


Scale factors on maps are generally given linearly. That means the scale factor is the scale comparing distances, not areas. So if 2 cm on the map represents 3 meters, you could write that as a fraction and convert it to a common unit of meters: 0.02 / 3, which you can multiply by 50 / 50 to get the fraction 1 / 150. That's the scale factor 1 : 150.

Almost always, when we talk about scale factors we mean the linear scale. So a toy car (in my day they were all called "dinky cars") like a Hot Wheels or Matchbox car is considered 1:64 scale, meaning that the toy's length is 1/64th the length of a real car. However, it is important to realize that this has big implications for the volume of the car. The scale of the toy car's volume is actually 1 : 262144. That's because volume equals distance times distance times distance. So the scale of the volume is 1/64 x 1/64 x 1/64.

Similarly, if you measure Levi's property on the map and see that it measures 800 cm2, how do we calculate his real property area? We can use the scale 2 cm = 3 m twice. 800 cm2 x (3m / 2 cm)2 = 7200 cm2 x m2 / 4 cm2 = 1800 m2. So he has 1800 square meters.

You see how I made the calculation work to cancel out the units I didn't want (any centimeters) and replace it with the proper real size units (meters).

The important things is to know that when a scale factor is given, unless otherwise stated explicitly, it is a linear scale factor, so if you are dealing with volumes or areas, you need to square or cube the scale to solve the problem.

Aside from that, it is just a fraction.

Stephen La Rocque

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