# Ratio and proportion

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

This entry interprets the numerical ratio a:b as a/b and defines a proportion as the equality of two valid ratios.

mko-ratio-proportion · definition · v0.10.0



## 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

[Ratio and proportion: solving 3:4=x:10, checking back, and testing data for direct proportion](https://ocme.evemisslab.com/en/derivations/derivation-mko-ratio-proportion-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Fraction](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/ratio-proportion.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-ratio-proportion/index.md)