6.5 Proportion Structure and Truth

Proportion Structure and Truth is the study of how two ratios are joined into a single equality statement called a proportion, how the terms of that statement correspond to one another, and how the truth or falsity of a proportion is verified numerically.

The Structure of a Proportion

Equality of Two Ratios

A proportion is a statement that two ratios are equal to one another. It asserts that the multiplicative relationship expressed by one ratio matches the multiplicative relationship expressed by another, even though the individual terms of the two ratios may differ.

Proportion Term Correspondence

A proportion has four terms, and each term occupies a specific position that corresponds to a position in the other ratio: the first and third terms are the numerators, the second and fourth are the denominators, and this correspondence must be preserved when reading, rearranging, or solving the proportion.

Proportion Reading and Interpretation

A proportion is read as "a is to b as c is to d," meaning that the relationship of a to b is the same kind of relationship that c has to d. This phrasing emphasizes that a proportion describes a shared pattern of comparison rather than a statement about the individual sizes of the four terms.

Determining Whether a Proportion Is True

True Numerical Proportion

A proportion is true when the two ratios it equates actually represent the same value once each is evaluated or reduced. The proportion 2 : 3 = 4 : 6 is true because both ratios reduce to the same simplified form.

False Numerical Proportion

A proportion is false when the two ratios do not reduce to the same value, even though the statement is written in valid proportion form. The statement 2 : 3 = 4 : 5 is a false proportion because the two ratios reduce to different values.

Verifying Proportion Truth

Proportion Cross Products

The cross products of a proportion are the two products formed by multiplying diagonally opposite terms: the first numerator by the second denominator, and the second numerator by the first denominator.

Cross-Product Equality

A proportion is true precisely when its two cross products are equal. This equality gives a purely numerical test for truth that does not require reducing either ratio to lowest terms, and is the standard method for confirming a proportion or solving for an unknown term within one.

Equivalent Fraction Interpretation

Since each ratio in a proportion can be written as a fraction, a true proportion is equivalent to a statement that two fractions represent the same value, which connects proportion truth directly to the concept of equivalent fractions and to cross-multiplication as a shared verification tool.

Proportions with Non-Whole-Number Terms

Proportion with Decimal Terms

When a proportion contains decimal terms, the cross-product test is applied identically: the diagonal products are computed using the decimal values, and the proportion is true exactly when those two decimal products match.

Proportion with Fractional Terms

When the terms of a proportion are themselves fractions, the cross-product test still applies, though computing each diagonal product requires multiplying fractions together, which may itself involve an intermediate common-denominator or numerator-times-numerator step before the two products can be compared.

Invalid Proportion from a Zero Denominator

A proportion is invalid, rather than simply true or false, if either denominator is zero, since a ratio with a zero denominator is undefined and cannot be meaningfully compared to another ratio. Any proportion containing such a term must be rejected before the cross-product test is applied, rather than evaluated as false.