4.1: Proportions

In our discussion of similar triangles the idea of a proportion will play an important role. In this section we will review the important properties of proportions.

A proportion is an equation which states that two fractions are equal. For example, \(\dfrac{2}{6}=\dfrac{4}{12}\) is a proportion. We sometimes say "2 is to 6 as 4 is to \(12\)." This is also written \(2: 6 = 4:12\). The extremes of this proportion are the numbers 2 and 12 and the means are the numbers 6 and 4. Notice that the product of the means \(6 \times 4=24\) is the same as the product of the extremes \(2 \times 12=24\).

EXAMPLES:

• \(\dfrac{2}{6} = \dfrac{4}{12}\) and \(2 \times 12=6 \times 4\) are both true.
• \(\dfrac{2}{3} = \dfrac{6}{9}\) and \(2 \times 9=3 \times 6\) are both true.
• \(\dfrac{1}{4} = \dfrac{4}{12}\) and \(1 \times 12=4 \times 4\) are both false.

Proof of Theorem 1: If \(\dfrac{a}{b} = \dfrac{c}{d}\), multiply both sides of the equation by \(bd\):

\[\dfrac{a}{\cancel{b}} (\cancel{b} d) = \dfrac{c}{\cancel{d}} (b \cancel{d})\]

We obtain \(ad = bc\).

Conversely, if \(ad = bc\), divide both sides of the equation by \(bd\):

\[\dfrac{d\cancel{d}}{b\cancel{d}} = \dfrac{\cancel{b}c}{\cancel{b}d}\]

The result is \(\dfrac{a}{b} = \dfrac{c}{d}\).

The following theorem shows that we can interchange the means or the extremes or both of them simultaneously and still have a valid proportion:

EXAMPLE: \(\dfrac{2}{6} = \dfrac{4}{12}, \dfrac{2}{4} = \dfrac{6}{12}, \dfrac{12}{6} = \dfrac{4}{2}, \dfrac{12}{4} = \dfrac{6}{2}\) are all true because \(2 \times 12=6 \times 4\).

The process of converting a proportion \(\dfrac{2}{6} = \dfrac{4}{12}\) to the equivalent equation \(2 \times 12 = 6 \times 4\) is sometimes called cross multiplication. The idea is conveyed by the following notation:

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