# Question 6961: True Fraction Sum versus Naive Representation Addition [Return to catalogue page 35](https://ocme.evemisslab.com/en/questions/page/35/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261011-advance-20261015-r01-fraction-foundations/#q-20261011-advance-20261015-r01-fraction-0011) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0012-06961 · q-20261011-advance-20261015-r01-fraction-0011 ## Original question and givens Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-2/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?). - G001 — Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-2/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?). ## Definition and scope 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. - [mko-fraction](https://ocme.evemisslab.com/en/mko/mko-fraction/) ## Step-by-step derivation S001 · impliesBefore original integer numerator/denominator representations;original denominators nonzero After representation and rational value are distinct data Conditions: none Negative denominators are valid,but reduced values do not replace original representations.Candidate algorithms about raw numerators/denominators must use those raw data.This finite audit is not a general proof of quotient well-definedness. Step identifiers and dependenciesRule: fraction_representation_value_scope · Uses: G001 S002 · calculationBefore true rational sum := sum(-1, 1/3) After true rational sum = -2/3 Conditions: none The true rational sum uses common-denominator addition,separate from naive representation addition. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore naive raw numerator sum := sum(-2, 1) After naive raw numerator sum = -1 Conditions: none The naive candidate adds the two ORIGINAL numerators,not reduced numerators. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore naive raw denominator sum := sum(2, 3) After naive raw denominator sum = 5 Conditions: none Likewise add the ORIGINAL denominators;each being nonzero does not make their sum nonzero. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore defined naive fraction candidate := div(-1, 5) After defined naive fraction candidate = -1/5 Conditions: none Only a nonzero original-denominator sum gives this candidate a rational value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore naive minus true sum := sub(-1/5, -2/3) After naive minus true sum = 7/15 Conditions: none Candidate minus true sum is the signed defect,not its absolute value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore raw numerator sum=-1;raw denominator sum=5;true sum=-2/3 After naive defined=Yes;naive=-1/5;correct=No Conditions: none Adding numerators and denominators separately is not fraction addition.Prior reduction may change the specified WRONG algorithm,so retain raw representations.An undefined candidate is not correct even when the true sum is0. Step identifiers and dependenciesRule: raw_fraction_naive_addition_audit · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute naive minus true sum with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute defined naive fraction candidate with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute naive raw denominator sum with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute naive raw numerator sum with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute true rational sum with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (-2/3, Yes, -1/5, 7/15, No) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 16319609231b876716c51bd34c73fc86bd99f4f728c312fadec5a5cb176e447c - candidate record SHA-256: 670f555b4f1fcad3721a44900a0dfe2544f6f91609f5921d549fcb1d689bf9c7 - semantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00 - question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261011-advance-20261015-r01-fraction-0011.json)