{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261022-r02-present-value-discounting-0007", "pilot_number": "R0058-29582", "number": 29582, "family_id": "qf-str22-presentvalue-one-period-discount", "title": { "zh-Hant": "第 29582 題：固定因子現值", "en": "Question 29582: One-Period Discount by a Positive Accumulation Factor" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261022-r02-present-value-discounting-0007", "batch_id": "batch-20261013-advance-20261022-r02-present-value-discounting-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-present-value-discounting" ], "title_zh": "固定因子現值", "stem_zh": "固定期率、固定期數且無現金流。操作=presentvalue-one-period-discount；資料={\"future\":[118,1],\"rate\":[-1,20]}。請依 [future,rate,factor,present] 欄序回答。", "answer": { "kind": "expression", "value": "(118, -1/20, 19/20, 2360/19)", "explanation_zh": "結構基礎核對=(118, -1/20, 19/20, 2360/19)；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；累積因子必須為正；不是投資建議或市場預測。" }, "difficulty": { "band": "upper_secondary", "level": 1, "skills": [ "固定因子現值" ] }, "variant": { "family_id": "qf-str22-presentvalue-one-period-discount", "seed": 202610222007, "parameters": { "future": [ 118, 1 ], "rate": [ -1, 20 ], "domain": "Exact_fixed_factor_discounting", "operation": "presentvalue-one-period-discount", "semantic_template": "qf-str22-presentvalue-one-period-discount/{\"future\":[118,1],\"rate\":[-1,20]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-13", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261013-advance-20261022-r02-present-value-discounting-0007", "batch_id": "batch-20261013-advance-20261022-r02-present-value-discounting-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "144397b38804fbcb1d172d783ef215cce4ce6fba326d9ec7a7dc64dd61578a21", "family_id": "qf-str22-presentvalue-one-period-discount", "seed": 202610222007, "semantic_template": "qf-str22-presentvalue-one-period-discount/{\"future\":[118,1],\"rate\":[-1,20]}" }, "fields": { "title": "One-Period Discount by a Positive Accumulation Factor", "stem": "Use a fixed rate i=-1/20 per declared period, with 1+i>0, fixed period conventions and no cash flows, fees, taxes or rounding. Amounts are nonnegative exact rationals. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. This is not investment advice, a quote or a market forecast. future amount=118/1. Give (future amount,rate,1+rate,present value=future/(1+rate)). A permitted negative rate may make present value exceed future value.", "answer": { "kind": "expression", "value": "(118, -1/20, 19/20, 2360/19)", "explanation": "Discount by division, not multiplication; a positive accumulation factor does not require a nonnegative rate. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (118, -1/20, 19/20, 2360/19)." }, "skills": [ "One-Period Discount by a Positive Accumulation Factor" ], "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": "固定期率、固定期數且無現金流。操作=presentvalue-one-period-discount；資料={\"future\":[118,1],\"rate\":[-1,20]}。請依 [future,rate,factor,present] 欄序回答。", "en": "Use a fixed rate i=-1/20 per declared period, with 1+i>0, fixed period conventions and no cash flows, fees, taxes or rounding. Amounts are nonnegative exact rationals. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. This is not investment advice, a quote or a market forecast. future amount=118/1. Give (future amount,rate,1+rate,present value=future/(1+rate)). A permitted negative rate may make present value exceed future value." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-str22-presentvalue-one-period-discount, presentvalue-one-period-discount]", "after": "derived = [[future, rate, factor, present]]", "relation": "implies", "rule_id": "R58_EXACT_source_interface_and_answer_ROLE", "conditions": [], "explanation": { "zh-Hant": "依本題銀行的keys、domain或frame、operation與輸出角色。STR22/App20 tuple與legacy scalar不同；分數是否保/1按原契約，不能用共同外觀改題。", "en": "Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。", "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [Present value rate, [-1, 20]]", "after": "derived = [-1/20]", "relation": "implies", "rule_id": "STR22_input_RAW_pair1e6_before_sign_gcd_normalization", "conditions": [], "explanation": { "zh-Hant": "原pair兩component各<=1e6且den非0，之後正分母約分；不先約分繞raw界。這個STR22 profile不可移到SAFE BigInt舊題。", "en": "Check BOTH raw pair components within1e6 and nonzero denominator BEFORE sign/gcd normalization.This STR22 profile must not be imposed on legacy SAFE/BigInt tasks." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。", "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Present value ORIGINAL positive accumulation factor := sum(1, -1/20)", "after": "Present value ORIGINAL positive accumulation factor = 19/20", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Present value ORIGINAL positive accumulation factor，不以小數近似替代。", "en": "Recompute Present value ORIGINAL positive accumulation factor with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "-1/20" ], "output": "19/20", "label": "Present value ORIGINAL positive accumulation factor" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [Present value ORIGINAL positive accumulation factor, sum, [1, -1/20]]", "after": "derived = [[1], [19, 20], 19/20]", "relation": "implies", "rule_id": "STR22_SOURCE_gcd_crosscancel_THEN_raw1e6_BEFORE_final_reduction", "conditions": [], "explanation": { "zh-Hant": "來源加法先分母gcd，乘法先cross-cancel；驗所得raw分子／分母各1e6後才最終約分。除法先驗reciprocal非零。沒有把本銀行改成其他bank的profile，也不放寬hold。", "en": "SOURCE addition first uses denominator gcd,and multiplication cross-cancels.Check BOTH resulting raw components within1e6 BEFORE final reduction.Division qualifies the nonzero reciprocal.Do not substitute another bank profile or loosen a source hold." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。", "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [-1/20, 19/20]", "after": "derived = [是]", "relation": "implies", "rule_id": "Present_value_FIXED_model_same_rate_period_NOcashflows", "conditions": [], "explanation": { "zh-Hant": "模型只用同一固定每期rate、同一期別、無cashflows/費稅/rounding。1+rate>0才可compound/discount；rate可負，這不是市場預測或投資建議。", "en": "Use the SAME fixed per-period rate/period convention without cash flows,fees,taxes or rounding.Require1+rate>0;the rate may be negative.This is not a market forecast or investment advice." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。", "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [Present value future, [118, 1]]", "after": "derived = [118]", "relation": "implies", "rule_id": "STR22_input_RAW_pair1e6_before_sign_gcd_normalization", "conditions": [], "explanation": { "zh-Hant": "原pair兩component各<=1e6且den非0，之後正分母約分；不先約分繞raw界。這個STR22 profile不可移到SAFE BigInt舊題。", "en": "Check BOTH raw pair components within1e6 and nonzero denominator BEFORE sign/gcd normalization.This STR22 profile must not be imposed on legacy SAFE/BigInt tasks." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。", "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Present value ORIGINAL one-period future over factor := div(118, 19/20)", "after": "Present value ORIGINAL one-period future over factor = 2360/19", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代值並保原操作數；額外教學運算不重套raw輸入界，只有明定的來源rational operation驗自己的profile。", "en": "Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Present value ORIGINAL one-period future over factor，不以小數近似替代。", "en": "Recompute Present value ORIGINAL one-period future over factor with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "118", "19/20" ], "output": "2360/19", "label": "Present value ORIGINAL one-period future over factor" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [Present value ORIGINAL one-period future over factor, div, [118, 19/20]]", "after": "derived = [[1, 1], [2360, 19], 2360/19]", "relation": "implies", "rule_id": "STR22_SOURCE_gcd_crosscancel_THEN_raw1e6_BEFORE_final_reduction", "conditions": [], "explanation": { "zh-Hant": "來源加法先分母gcd，乘法先cross-cancel；驗所得raw分子／分母各1e6後才最終約分。除法先驗reciprocal非零。沒有把本銀行改成其他bank的profile，也不放寬hold。", "en": "SOURCE addition first uses denominator gcd,and multiplication cross-cancels.Check BOTH resulting raw components within1e6 BEFORE final reduction.Division qualifies the nonzero reciprocal.Do not substitute another bank profile or loosen a source hold." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。", "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [118, -1/20, 19/20]", "after": "derived = [2360/19]", "relation": "implies", "rule_id": "Present_value_ONE_period_inverse_of_same_forward_factor", "conditions": [], "explanation": { "zh-Hant": "由future=present*(1+rate)且factor正，兩邊除同一factor得present；不是再乘factor。負rate可使present大於future。", "en": "From future=present*(1+rate) and positive factor,divide by the SAME factor to recover present,not multiply again.A negative rate may make present exceed future." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原題資料、同一索引／方向／代表式與本步前提；不只核答案。精確候選核對、獨立審查、形式證明與發布資格仍分開。", "en": "Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null } ], "math_model": { "result": [ "118", "-1/20", "19/20", "2360/19" ] }, "conclusion": { "canonical_value": "(118, -1/20, 19/20, 2360/19)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261013-advance-20261022-r02-present-value-discounting-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261022-r02-present-value-discounting-foundations/questions.jsonl", "batch_sha256": "e2c003636a294a379610980b95a2281469ac42351f8bd9754db9411f8eb096f4", "original_line": 7, "original_line_sha256": "429e4abfc380c818d3470b9690cddfeccad2bc5a73a40862f5cb56dc4ad77e86", "question_compact_sha256": "144397b38804fbcb1d172d783ef215cce4ce6fba326d9ec7a7dc64dd61578a21", "english_row_compact_sha256": "94e7e193e7de4e93216326b07eaa9b44758f0dd3f435c46a4953b332ebbedd1b", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261022-r02-present-value-discounting-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261022-r02-present-value-discounting-foundations.en.jsonl" }, "mko_bindings": { "mko-present-value-discounting": { "sha256": "2e47f9d121504d577795a46f4235f98114b5701015039cc83bef029cead220b0", "english_sha256": "209445c349c4cc3d484122872870e437c5723eb7e12b40900d44b653eb99a043", "original_path": "public/data/mko/present-value-discounting.json", "english_path": "public/data/i18n/editions/mko-present-value-discounting.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "623c23f45586b08629c5a347ffbc994cf8dbff2133c4f4cdb227b851d74ea6b6", "original_candidate_sha256": "c86e603f27ccceb4ebfe9af30f328208af7fc760d09bca321dde5cf2ec57ccef", "original_candidate_line_sha256": "c86e603f27ccceb4ebfe9af30f328208af7fc760d09bca321dde5cf2ec57ccef" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-011-p59", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05df", "question_first_witness_sha256": "b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 9, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "b4978e9f3a9240677012a81510b41e8eafb9cdcf68d981c65ecaaa35b86a1a03", "arithmetic_steps": 2 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }