How Does Scale Ratio Work?

Scale is a tool you use when you are looking at maps, for example. For a map to have any purpose, it has to be scaled down. It needs to fit on a sheet of paper or on the screen of your PC.

A scale is written like this: $1:10\phantom{\rule{0.17em}{0ex}}000$. This means that $1$ cm on the map or drawing will correspond to $10\phantom{\rule{0.17em}{0ex}}000$ cm in real life. So the first number says something about what you measure on the map, while the second number tells you how much it is in reality.

Scale can work the other way as well, to make things bigger. For example, we can write the scale like this: $100:1$. Then, $100$ cm on the drawing will correspond to $1$ cm in reality. Can you imagine anything you can draw blueprints of? It could be the components inside an iPhone, a room in a house, or a car.

Example 1

There is $\text{}6\text{}\phantom{\rule{0.17em}{0ex}}\text{cm}$ between Bergen and Oslo on a map. In reality the distance between the two cities is $\text{}300\text{}\phantom{\rule{0.17em}{0ex}}\text{km}$. What is the scale of this map?

First, you have to find how many centimeters are in $300$ km: $\begin{array}{llll}\hfill 300\phantom{\rule{0.17em}{0ex}}\text{km}& =\left(300\cdot 1000\right)\phantom{\rule{0.33em}{0ex}}\text{m}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =300\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}\text{m}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\left(300\phantom{\rule{0.17em}{0ex}}000\cdot 100\right)\phantom{\rule{0.33em}{0ex}}\text{cm}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =30\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}\text{cm}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

This means $6$ cm on the map corresponds to $30\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000$ cm in reality. Then you can find the scale like this: $\begin{array}{llll}\hfill 6& ÷30\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \frac{6}{6}& ÷\frac{30\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000}{6}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 1& ÷5\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

The scale is $1:5\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000$.

Example 2

A globe has a radius of $\text{}20\text{}\phantom{\rule{0.17em}{0ex}}\text{cm}$. The radius of the earth is approximately $\text{}6371\text{}\phantom{\rule{0.17em}{0ex}}\text{km}$. What is the scale of the globe?

First, you need to find how many centimeters are in $6371$ km: $\begin{array}{llll}\hfill 6371\phantom{\rule{0.17em}{0ex}}\text{km}& =\left(6371\cdot 1000\right)\phantom{\rule{0.33em}{0ex}}\text{m}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =6\phantom{\rule{0.17em}{0ex}}371\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}\text{m}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\left(6\phantom{\rule{0.17em}{0ex}}371\phantom{\rule{0.17em}{0ex}}000\cdot 100\right)\phantom{\rule{0.33em}{0ex}}\text{cm}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =637\phantom{\rule{0.17em}{0ex}}100\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}\text{cm}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

That means $20$ cm on the globe is equivalent to $637\phantom{\rule{0.17em}{0ex}}100\phantom{\rule{0.17em}{0ex}}000$ cm in reality. Then you can find the scale like this: $\begin{array}{llll}\hfill 20& ÷637\phantom{\rule{0.17em}{0ex}}100\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \frac{20}{20}& ÷\frac{637\phantom{\rule{0.17em}{0ex}}100\phantom{\rule{0.17em}{0ex}}000}{20}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 1& ÷31\phantom{\rule{0.17em}{0ex}}855\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

The scale is $1:31\phantom{\rule{0.17em}{0ex}}855\phantom{\rule{0.17em}{0ex}}000$.