Proportions

In mathematics, a proportion is a relation of equality between two ratios or rates.

To form a proportion, the two ratios or rates must be equivalent.

The following ratios are proportional: ||\displaystyle 3:4=15:20||
Thus, the two ratios are equivalent. ||\begin{align}3\div 4&=0.75\\ 15\div20&=0.75\end{align}||

The following rates are proportional: ||\displaystyle \frac{300\ \text{inhabitants}}{5\ \text{km}^2}=\frac{600\ \text{inhabitants}}{10\ \text{km}^2}|| Thus, the two rates are equivalent. ||\begin{align}300\div 5&=60\ \text{inhabitants/km}^2\\ 600\div 10&=60\ \text{inhabitants/km}^2\end{align}||

In a proportion, the first and fourth terms are called the extremes of the proportion. The second and third terms are called the means of the proportion. ||\displaystyle \frac{\text{Extreme}}{\text{Mean}}=\frac{\text{Mean}}{\text{Extreme}}|| In other words, in the proportion ||\color{blue}{a}:\color{green}{b}=\color{green}{c}:\color{blue}{d}\\ \text{or}\\ \displaystyle \frac{\color{blue}{a}}{\color{green}{b}}=\frac{\color{green}{c}}{\color{blue}{d}}|| terms |\color{blue}{a}| and |\color{blue}{d}| are the extremes and terms |\color{green}{b}| and |\color{green}{c}| are the means.

Consider the following proportion: ||\displaystyle \frac{\color{blue}{1}}{\color{green}{2}}=\frac{\color{green}{4}}{\color{blue}{8}}|| Terms |\color{blue}{1}| and |\color{blue}{8}| are the extremes.
Terms |\color{green}{2}| and |\color{green}{4}| are the means.

In a proportion, the product of the extremes is equal to the product of the means.

If |\displaystyle \frac{\color{blue}{a}}{\color{green}{b}}=\frac{\color{green}{c}}{\color{blue}{d}}|,

thus |\color{blue}{a}\times \color{blue}{d}=\color{green}{b}\times \color{green}{c}|.

Consider the following proportion: ||\displaystyle \frac{3}{4}=\frac{9}{12}|| Note the product of the extremes is equal to the product of the means. ||\begin{align}3\times 12&=4\times 9\phantom{1}\\36&=36\end{align}||

The proportionality ratio is the number by which to multiply the numerator of a proportion’s rates or ratios to obtain the denominator.

Consider the following proportion: ||\displaystyle \frac{2}{6}=\frac{7}{21}|| In this proportion, the proportionality ratio is |\color{red}{3}|.

In a proportion, the factor of change is the number by which to multiply the numerator (or denominator) of one ratio or rate to obtain the numerator (or denominator) of the other ratio or rate.

Consider the following proportion: ||\displaystyle \frac{4}{5}=\frac{24}{30}||

In this proportion, the factor of change is |\color{red}{6}|.