Chapter 23: Ratio and Proportion

A Ratio is the relation which one quantity bears to another of the same kind in respect to magnitude. A Proportion is an expression of equality between two ratios. While these concepts are introduced in basic arithmetic, Algebra allows us to manipulate them as literal equations.

23.1 Ratio

The ratio of a to b is written as a : b or as the fraction a/b. The first term is called the antecedent and the second is the consequent.

A ratio is not changed if both terms are multiplied or divided by the same number.
Ratios are compared by reducing them to a common denominator.

Example: Which is the greater ratio, 5:7 or 15:22?

Convert to fractions: 5/7 and 15/22.
Common denominator (154):
5/7 = 110/154
15/22 = 105/154

Result: 5:7 is the greater ratio.

23.2 Proportion

A proportion is an equation stating that two ratios are equal. It is written as a : b = c : d or a/b = c/d. The terms a and d are called the extremes, while b and c are the means.

The Fundamental Property: In every proportion, the product of the extremes is equal to the product of the means.
If a/b = c/d, then ad = bc.

23.3 Transformation of Proportions

If a/b = c/d, the following must also be true:

Inversion: b/a = d/c
Alternation: a/c = b/d
Composition: (a+b)/b = (c+d)/d
Division: (a-b)/b = (c-d)/d

Doctor's Pro-Tip: Composition and Division are incredibly useful when solving complex geometric problems or working with chemical mixtures where the total volume changes but the ratio of ingredients stays the same.