Simplify a ratio a : b — divide both sides by the greatest common divisor (GCD). If using decimals, scale to integers first, then reduce.
Equivalent ratios — multiply or divide both sides by the same non-zero number (e.g., 4:3 → 8:6 → 12:9).
Proportion — if a : b = c : d and one term is missing, use cross-multiplication: A×D = B×C.
Scale factor — k ≈ C/A ≈ D/B (when A,B ≠ 0). Use k to scale between equivalent ratios.
Shares of total — For a+b ≠ 0, left share = a/(a+b), right share = b/(a+b).
Edge cases — 0 : b reduces to 0 : 1; a : 0 reduces to 1 : 0; 0 : 0 is undefined.
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