{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20260930-r01-fraction-exact-addition-0020", "pilot_number": "R0063-32145", "number": 32145, "family_id": "qf-fraction-exact-addition", "title": { "zh-Hant": "第 32145 題：分數精確加法", "en": "Question 32145: Exact Fraction Addition" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20260930-r01-fraction-exact-addition-0020", "batch_id": "batch-20260930-r01-fraction-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-fraction" ], "title_zh": "分數精確加法", "stem_zh": "對整數構成的合法分數，精確計算 7/(-6)+(5)/(5)，寫成最簡正分母 p/q。", "answer": { "kind": "short_text", "value": "-1/6", "explanation_zh": "通分的分子為 5，分母為 -30；正規化並約分得到 -1/6。" }, "difficulty": { "band": "lower_secondary", "level": 2, "skills": [ "通分", "精確有理數" ] }, "variant": { "family_id": "qf-fraction-exact-addition", "seed": 202609301320, "parameters": { "operation": "addition", "p": 7, "q": -6, "r": 5, "s": 5, "semantic_template": "qf-fraction-exact-addition/{\"operation\":\"addition\",\"p\":7,\"q\":-6,\"r\":5,\"s\":5}" } }, "verification": { "status": "candidate", "checks": [ "json_schema", "producer_local_answer_invariants", "explanation_witness_accuracy" ], "warning_zh": "本題保持 candidate；有限精確計算與產生者自檢不構成人工審查、形式證明或主架構者驗收。" }, "provenance": { "generator_role": "question_producer", "generator_model": "Codex GPT-6 question producer", "generated_at": "2026-09-30", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20260930-r01-fraction-exact-addition-0020", "batch_id": "batch-20260930-r01-fraction-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "9987d3ab3f493880057401b9df7c3e51092460d8cea22d8fef3d4b60d6fe0b0f", "family_id": "qf-fraction-exact-addition", "seed": 202609301320, "semantic_template": "qf-fraction-exact-addition/{\"operation\":\"addition\",\"p\":7,\"q\":-6,\"r\":5,\"s\":5}" }, "fields": { "title": "Exact Fraction Addition", "stem": "Compute 7/(-6)+(5)/(5) exactly and write the result in lowest terms with a positive denominator.", "answer": { "kind": "short_text", "value": "-1/6", "explanation": "Use the common-denominator numerator 7×5+(5)×-6 and denominator -6×5, then normalize and reduce to -1/6." }, "skills": [ "common denominators", "exact rational arithmetic" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "對整數構成的合法分數，精確計算 7/(-6)+(5)/(5)，寫成最簡正分母 p/q。", "en": "Compute 7/(-6)+(5)/(5) exactly and write the result in lowest terms with a positive denominator." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-fraction-exact-addition, addition, no-required-domain, []]", "after": "derived = [[reduced_fraction_RETAIN1_STRING]]", "relation": "implies", "rule_id": "R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle", "conditions": [], "explanation": { "zh-Hant": "BASE來源沒有本輪legacy的通用operation/domain/額外欄位禁令；保metadata而不當答案。PULSE與legacy則核來源精確欄位、operation及domain。不要把tuple、/1或輸入界強套全部題族。", "en": "BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。", "en": "Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [7, -6, [5, 5]]", "after": "derived = [是]", "relation": "implies", "rule_id": "Fraction_ALL_RAW_SAFE_integer_components_even_zero_den_validity", "conditions": [], "explanation": { "zh-Hant": "先驗全部raw分量的SAFE整數型別，包含q0／0/0資格題的分子。資格題可合法No；其他實際運算才要求所有用到的den非0。", "en": "Validate EVERY raw SAFE integer component,including the numerator at q0/0/0 eligibility.A validity task may legally return No;actual arithmetic requires ALL used denominators nonzero." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。", "en": "Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Fraction ORIGINAL p times s := mul(7, 5)", "after": "Fraction ORIGINAL p times s = 35", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Fraction ORIGINAL p times s，不以小數近似替代。", "en": "Recompute Fraction ORIGINAL p times s with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7", "5" ], "output": "35", "label": "Fraction ORIGINAL p times s" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Fraction ORIGINAL r times q := mul(5, -6)", "after": "Fraction ORIGINAL r times q = -30", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Fraction ORIGINAL r times q，不以小數近似替代。", "en": "Recompute Fraction ORIGINAL r times q with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "5", "-6" ], "output": "-30", "label": "Fraction ORIGINAL r times q" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Fraction addition ORIGINAL cross numerator sum := sum(35, -30)", "after": "Fraction addition ORIGINAL cross numerator sum = 5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Fraction addition ORIGINAL cross numerator sum，不以小數近似替代。", "en": "Recompute Fraction addition ORIGINAL cross numerator sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "35", "-30" ], "output": "5", "label": "Fraction addition ORIGINAL cross numerator sum" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Fraction addition ORIGINAL denominator product := mul(-6, 5)", "after": "Fraction addition ORIGINAL denominator product = -30", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Fraction addition ORIGINAL denominator product，不以小數近似替代。", "en": "Recompute Fraction addition ORIGINAL denominator product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-6", "5" ], "output": "-30", "label": "Fraction addition ORIGINAL denominator product" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [7, -6, 5, 5]", "after": "derived = [5, -30]", "relation": "implies", "rule_id": "Fraction_ADDITION_common_den_ALL_cross_terms_not_naive_components", "conditions": [], "explanation": { "zh-Hant": "通分後分子ps+rq，分母qs；兩個交叉項都保留，不是分子直接加／分母直接加。衍生大整數不限SAFE。", "en": "Use the common denominator qs with numerator ps+rq,retaining BOTH cross terms.This is not componentwise addition.Derived large integers are not SAFE-capped." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。", "en": "Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Fraction addition normalization ORIGINAL negate numerator for denominator sign := sub(0, 5)", "after": "Fraction addition normalization ORIGINAL negate numerator for denominator sign = -5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Fraction addition normalization ORIGINAL negate numerator for denominator sign，不以小數近似替代。", "en": "Recompute Fraction addition normalization ORIGINAL negate numerator for denominator sign with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "5" ], "output": "-5", "label": "Fraction addition normalization ORIGINAL negate numerator for denominator sign" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Fraction addition normalization ORIGINAL negate denominator := sub(0, -30)", "after": "Fraction addition normalization ORIGINAL negate denominator = 30", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Fraction addition normalization ORIGINAL negate denominator，不以小數近似替代。", "en": "Recompute Fraction addition normalization ORIGINAL negate denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "-30" ], "output": "30", "label": "Fraction addition normalization ORIGINAL negate denominator" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [Fraction addition normalization, 5, -30]", "after": "derived = [-5, 30]", "relation": "implies", "rule_id": "Fraction_DEN_nonzero_sign_normalized_BEFORE_gcd", "conditions": [], "explanation": { "zh-Hant": "分母先非0才可表示分數；若負則分子分母同乘−1。衍生num/den不套raw SAFE；不只改分母而改掉值。", "en": "A denominator must FIRST be nonzero.If negative,negate numerator AND denominator together.Derived numerator/denominator are not RAW-SAFE recapped;changing only the denominator would change the value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。", "en": "Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Fraction addition normalization gcd step0 ORIGINAL quotient times divisor := mul(0, 30)", "after": "Fraction addition normalization gcd step0 ORIGINAL quotient times divisor = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Fraction addition normalization gcd step0 ORIGINAL quotient times divisor，不以小數近似替代。", "en": "Recompute Fraction addition normalization gcd step0 ORIGINAL quotient times divisor with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "30" ], "output": "0", "label": "Fraction addition normalization gcd step0 ORIGINAL quotient times divisor" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Fraction addition normalization gcd step0 ORIGINAL dividend minus product := sub(5, 0)", "after": "Fraction addition normalization gcd step0 ORIGINAL dividend minus product = 5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Fraction addition normalization gcd step0 ORIGINAL dividend minus product，不以小數近似替代。", "en": "Recompute Fraction addition normalization gcd step0 ORIGINAL dividend minus product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "5", "0" ], "output": "5", "label": "Fraction addition normalization gcd step0 ORIGINAL dividend minus product" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "source roles = [Fraction addition normalization, 0, 5, 30, 0]", "after": "derived = [0, 5]", "relation": "implies", "rule_id": "Fraction_EUCLID_all_remainders_and_common_divisor_invariant", "conditions": [], "explanation": { "zh-Hant": "以x=qy+r、0≤r