6.3 Introduction and Key Insights

You have probably heard and used the words proportion, proportional, and disproportionate. How about, "When that hungry math professor was told she could not purchase 49 cannolis, screaming and throwing items seemed like a disproportionate response"? Mathematically speaking, a proportion is simply an equation that relates two ratios.

For example, the equation $\frac{30}{9} = \frac{10}{3}$ is an example of a proportion. Ancient Greek mathematicians seriously loved proportions, especially when they applied them to similar shapes!

Let's look at these two rectangles. The one on the right was obtained by copy-pasting the first one on the left and then resizing by a factor of $1/2$.

When comparing the corresponding side lengths for these two rectangles, we find proportions. In particular, we find:

$\frac{\text{Big Rectangle's Height}}{\text{Small Rectangle's Height}} = \frac{2}{1}= \frac{\text{Big Rectangle's Width}}{\text{Small Rectangle's Width}}.$

It is interesting that ancient mathematicians always viewed fractions and the equations that involved them as statements about ratios and proportions. For example, the expression $1/3$ did not refer to a number or a quantity, such as "one third of the way between zero and one". It was necessarily and exclusively a comparison of two measurements.

Manipulating proportions involves the multiplication and division skills we mastered in Unit 4, where we practiced multiplying and dividing fractions.