Proportions
The concept of proportion is closely related to the concept of ratio; it is nearly impossible to discuss one without defining the other. Specifically, a proportion is an equation of two equal ratios; if two (or more) ratios are equal, they are considered proportional. The ratios 3/5, 6/10, and 12/20, for example, are proportional; each is a constant multiple of another. If each element of the ratio is a constant multiple of the associated element in the second ratio, then the ratios are considered proportional. The ratio 1:3:8, for example, is proportional to the ratio 4:12:32, since each element in the second ratio is exactly the associated element in the first ratio multiplied by 4. However, the ratio 1:3:8 is not proportional to the ratio 4:12:26. In shorthand notation, proportions may be written as x:y::ab, which reads "x is to y as a is to b."
Deriving the basic proportion equation—a/b = c/d--involves basic algebra. It is based on the basic relationship of setting two products equal to each other, such as a*d = b*c. This equality implies that a = (b*c)/d and finally that a/b = c/d. To return to the original equation, the mathematical shortcut of cross-multiplication is used in which the numerator of each ratio (e.g., a) is multiplied by the denominator of the other ratio (e.g., d). Both products are then set equal.
Proportions are applied often to everyday events and situations; it is one of the most widely applied concepts from mathematics. A percentage, which represents the number of events out of 100, is determined by setting a known ratio a/bequal to the ratio x/100. Then, a is said to be "x%" of b. Suppose, for example, that 26 of 40 participants in a life-saving certification class at college A were women, and 64 of 160 participants in a life-saving certification class at college B were women. Based on the concept of proportion and percentage, one could equivalently say that "65% of the 40 participants at college A and 40% of the 160 participants at college B were women." Since the total number of participants is known at each college, it is also a matter of simple algebra to determine the actual number of women participants at each class.
Since percentages are ratios that are normalized to a common and well-understood standard (that is, normalized to 100), they are a good tool for comparing values when used properly. Relative to the total number of participants at each college, a greater percentage of the class were women at college A than college B (65% is greater than 40%). Notice, however, that percentages can also be somewhat misleading. Although 65% of the participants at college A were women and only 40% of the participants at college B were women, a greater number of women actually attended the college B session (64 at college B vs. 26 at college A). Comparison of percentages must always be considered in their proper context.
The concept of similarity is also based on proportion. Similar triangles, for example, are triangles in which each set of associated angles are equal--that is, if triangle ABC is similar to triangle DEF, then the magnitude of angle A equals the magnitude of angle D, the magnitude of angle B equals the magnitude of angle E, and the magnitude of angle C equals the magnitude of angle F. As a result of the similarity, the ratios of the associated line segments is are equal--AB:DE::BC:EF::CA:FD. This idea can be extended to any set of similar polygons and is fundamentally important axiom in geometry.
While proportion is primarily a mathematical concept—an equality of ratios—it also has aesthetic value in both the natural and man-made environments and expressions of these environments. Artistic caricatures, for example, tend to exaggerate some element of the subject to excessive proportion in order to draw attention to it, such as an overwhelmingly large hat or unfeasibly small car. The viewer notices the exaggeration precisely because the artist has inserted an unexpected proportionality into the artwork, relative to the other objects in the piece. The hat appears too large for the subject--or the car appears too small--because other objects in the caricature are sized in a normal range relative to each other. Art that captures proportion accurately is often considered quite striking; art that exaggerates proportion may even be called grotesque. Beyond its mathematical implications, then, the human experience with proportion affects its aesthetic interpretation.
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