{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-ratio-proportion"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-ratio-proportion-01","title":{"zh-Hant":"比與比例：解 3:4=x:10、代回核對與檢查正比例資料","en":"Ratio and proportion: solving 3:4=x:10, checking back, and testing data for direct proportion"},"summary":{"zh-Hant":"在有理數中：(1) 證明對 b≠0、d≠0，比例 a:b=x:d 恰有一個有理解 x=ad/b；(2) 用它解 3:4=x:10 並代回核對；(3) 說明 4:3=x:10 的解不同，比的順序有意義；(4) 判定資料 (2,5)、(4,10)、(6,15)、(8,21) 是否符合同一個正比例 y=kx。","en":"Over the rationals: (1) prove that for b≠0 and d≠0 the proportion a:b=x:d has exactly one rational solution, x=ad/b; (2) use it to solve 3:4=x:10 and check back; (3) show that 4:3=x:10 has a different solution, so the order of a ratio matters; (4) decide whether the data (2,5), (4,10), (6,15), (8,21) fit a single direct proportion y=kx."},"mko_ids":["mko-ratio-proportion"]}],"title":"Concept derivation guides"}