1.6 Ratio and Proportion Definitions

Ratio and Proportion Definitions establishes the vocabulary used to compare quantities multiplicatively rather than additively, covering the terms needed to describe a single comparison between two quantities, a comparison involving different units, and the equality between two such comparisons that allows unknown quantities to be solved for.

Ratio and Ratio Order

A ratio is a comparison of two quantities by division, expressing how many times one quantity contains, or is contained within, another, commonly written as a to b, a:b, or a/b. Ratio order refers to the fixed sequence in which the two quantities are compared; the ratio of a to b is not the same as the ratio of b to a unless the two quantities are equal, so the order in which terms are written carries essential meaning.

Equivalent Ratios

Equivalent ratios are two or more ratios that express the same underlying comparison despite using different numbers, obtained by multiplying or dividing both terms of a ratio by the same nonzero number. The ratio 2:3 is equivalent to 4:6 and to 10:15, since each preserves the same relative relationship between the two quantities.

Rate and Unit Rate

A rate is a ratio that compares two quantities measured in different units, such as distance compared to time. A unit rate is a rate expressed with a denominator of exactly 1, describing how much of one quantity corresponds to a single unit of the other, such as 60 miles per 1 hour. Unit rates are the standard form used when comparing rates across different situations, since reducing every rate to the same denominator of 1 makes direct comparison straightforward.

Compound Units

A compound unit is a unit of measurement formed by combining two or more simpler units through multiplication or division, such as miles per hour (distance divided by time) or dollars per square foot (cost divided by area). Compound units arise naturally from rates, since a rate expresses one quantity per unit of another, and the resulting combined unit must be carried through any subsequent calculation.

Conversion Factor

A conversion factor is a ratio, equal in value to 1, used to convert a quantity from one unit of measurement into an equivalent quantity expressed in a different unit, such as the ratio of 12 inches to 1 foot. Because a conversion factor equals 1, multiplying any quantity by it changes the unit of measurement without changing the underlying value being measured.

Proportion and Proportional Quantities

A proportion is a statement of equality between two ratios, asserting that the ratio of a to b equals the ratio of c to d. Proportional quantities are two sets of related values that maintain a constant ratio to one another across every corresponding pair, meaning as one quantity increases or decreases, the other changes by the same multiplicative factor, keeping the ratio between them fixed.

Scale Factor

A scale factor is the constant ratio by which every dimension of an original figure or quantity is multiplied to produce a proportionally resized version of it, such as a scale factor of 2 doubling every length in a figure while preserving its overall shape. Scale factors provide the mechanism through which proportional relationships are applied concretely to resize maps, models, drawings, or any set of proportionally linked measurements.

Together, these definitions establish the multiplicative comparison framework used throughout elementary algebra: ratios describe a single comparison, rates and unit rates extend that comparison across different units, conversion factors and compound units carry those comparisons through unit changes, and proportions and scale factors formalize the equality between two ratios that underlies solving for unknown proportional quantities.