# Question 6483: Missing Rational Addend and Substitution Defect [Return to catalogue page 33](https://ocme.evemisslab.com/en/questions/page/33/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261009-advance-20261013-r03-rational-number-foundations/#q-20261009-advance-20261013-r03-rational-number-0083) 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. R0011-06483 · q-20261009-advance-20261013-r03-rational-number-0083 ## Original question and givens Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 9/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers. - G001 — Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 9/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers. ## 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-rational-number](https://ocme.evemisslab.com/en/mko/mko-rational-number/) ## Step-by-step derivation S001 · impliesBefore raw a=[1, 2];raw b=[-1, 3] After a=1/2;b=-1/3 Conditions: none Original denominators need only be nonzero;negative denominators are valid.Normalize to reduced positive-denominator fractions for exact arithmetic while retaining the original representations and sources. Step identifiers and dependenciesRule: nonzero_signed_rational_representation · Uses: G001 S002 · calculationBefore target minus a := sub(-3/2, 1/2) After target minus a = -2 Conditions: none From x+a+b=target,subtract a first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore missing rational addend := sub(-2, -1/3) After missing rational addend = -5/3 Conditions: none Then subtract b to obtain the unique rational addend x. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore reverse x+a := sum(-5/3, 1/2) After reverse x+a = -7/6 Conditions: none Substitute the recovered x back and add a. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore reverse x+a+b := sum(-7/6, -1/3) After reverse x+a+b = -3/2 Conditions: none Add the original b to reconstruct the original left side. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore missing addend substitution residual := sub(-3/2, -3/2) After missing addend substitution residual = 0 Conditions: none Subtract the original target from the substituted value;the result must be exactly0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore x=-5/3;target=-3/2 After substitution residual=0/1 Conditions: none Audit the rearrangement by full substitution,not by copying the target or confusing zero residual with undefinedness. Step identifiers and dependenciesRule: rational_missing_addend_reverse_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 missing addend substitution residual with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute reverse x+a+b with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute reverse x+a with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute missing rational addend with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute target minus a 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 (-5/3, 0/1) 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: 31fc0a252c51eb54f09aefc15ad55874a9ee42238ad030093133c35866b5b574 - candidate record SHA-256: f4113581e0bbb4f73e5238712453efbe7c774466a88ed9930fda6f1b39ac1ebf - semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80 - question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261009-advance-20261013-r03-rational-number-0083.json)