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R0011-06483 · q-20261009-advance-20261013-r03-rational-number-0083
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.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.
Before
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.
Rule: nonzero_signed_rational_representation · Uses: G001
Before
target minus a := sub(-3/2, 1/2)
After
target minus a = -2
Conditions: none
From x+a+b=target,subtract a first.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
reverse x+a := sum(-5/3, 1/2)
After
reverse x+a = -7/6
Conditions: none
Substitute the recovered x back and add a.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
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.
Rule: rational_missing_addend_reverse_audit · Uses: G001, S006
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.(-5/3, 0/1)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 31fc0a252c51eb54f09aefc15ad55874a9ee42238ad030093133c35866b5b574candidate record SHA-256: f4113581e0bbb4f73e5238712453efbe7c774466a88ed9930fda6f1b39ac1ebfsemantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55Lean: NOT_RUN · human review: NOT_PERFORMED