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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, Math Kangaroo 5-6, Math Kangaroo 7-8.

A ratio is a multiplicative comparison of two quantities.

Numerically, a ratio is the same number as a fraction, however, conceptually, ratios embody a different meaning. Ratios apply to comparing quantities that have the same physical nature. We cannot compare mass with volume, therefore there is no such thing as a ratio of 'mass to volume'. But we can definitely define a new quantity such as density, which is the result of dividing the mass of an object by its volume - this quantity is not a comparison, therefore the division is not a 'ratio'.

For example, we can say: "the population of San Francisco and the population of Ankara are in a ratio of 1:5." But it does not make much sense to say: "the population of Paris is in a ratio of 112,000:31 to the height of the Tour Eiffel."

To compare quantities by means of a ratio, they have to have the same physical nature and they have to be measured with the same unit.

Therefore, in writing a ratio, it may be necessary to convert units.

The ratio is a dimensionless quantity, since the units simplify. Example:

By contrast, a fraction can represent a physical quantity measured in some kind of unit. For example:

Property: A ratio is generally not commutative.

Property: Both terms of a ratio may be multiplied by the same factor to obtain an equivalent ratio:

Definition: A proportion is a pair of equivalent ratios.

Definition: outer terms (or extremes) of a proportion

Definition: inner terms (or means) of a proportion

Property (cross-multiplication): the product of the inner terms is equal to the product of the outer terms

Property: the inner terms may be interchanged

Property: the outer terms may be interchanged

Property: in a multiple ratio, the ratio formed by adding all the numerators and all the denominators is also part of the set

Property: in a proportion we can add/subtract the denominators from the numerators and the result is a proportion

Property: the denominators and the numerators can be interchanged

All the above properties can easily be proven using elementary algebra.

Definition: Four numbers are proportional if they can be assigned to the terms of a proportion (i.e. two equivalent ratios can be formed with them).