Ratio and proportion

Mathematical statement

For rational numbers a, b, c, and d with b≠0 and d≠0, the value of the numerical ratio a:b is a/b. The proportion a:b=c:d means a/b=c/d and is equivalent to ad=bc.

Representative formula (exact TeX)

\frac{a}{b}=\frac{c}{d}

Assumptions

• This entry uses numerical ratios over the rational numbers, with b and d nonzero.
• When ratios describe quantities, their units and comparability are stated first; numerical proportionality alone does not establish geometric similarity or causation.

Symbol scope

• a — antecedent of the first ratio
• b — nonzero consequent of the first ratio
• c — antecedent of the second ratio
• d — nonzero consequent of the second ratio

Intuition, definition, and misconceptions

Order matters in a ratio. In general a:b differs from b:a, and reversing the ratio additionally requires a≠0.

Before two ratios form a proportion, verify that both consequents are nonzero and then compare the cross products ad and bc.

When b and d are nonzero, the proportion a/b=x/d has the unique rational solution x=ad/b. Finite data for direct variation y=kx must check the same k at every listed point; one point does not justify a claim about every input.

Equal cross products do not make a zero-denominator expression valid, and equal numerical ratios do not automatically establish geometric similarity.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Fraction — Numerical ratios and proportions use valid fraction representations and cross products.

Verification boundary

Exact ratio, proportion, and finite-point checks cover only the stated conditions. General application models and geometric similarity have separate proof obligations.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind rational division by a nonzero value, the cross-product equivalence, and missing-term proportion solving.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.