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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).