# Question 6993: Two Raw Scaled Fraction Representations [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-0093) 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. R0012-06993 · q-20261011-advance-20261015-r01-fraction-0093 ## Original question and givens Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=5, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic. - G001 — Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=5, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic. ## 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 original fraction value := div(5, -3) After original fraction value = -5/3 Conditions: none With nonzero original denominator,compute the original value as the audit anchor for nonzero scaling. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore scaled numerator0 := mul(5, 2) After scaled numerator0 = 10 Conditions: none Multiply the original numerator by this integer scale,retaining the exact integer product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore scaled denominator0 := mul(-3, 2) After scaled denominator0 = -6 Conditions: none Multiply the original denominator by the same scale,without prior reduction or deletion of zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore scaled rational value0 := div(10, -6) After scaled rational value0 = -5/3 Conditions: none Reduce to a rational value only when the new denominator is nonzero. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore scale=2;raw scaled pair=[10, -6] After legal=Yes;normalized value=-5/3 Conditions: none A nonzero scale cancels and preserves value;zero scaling produces0/0,which is invalid even with zero numerator and cannot be reduced back to the original value. Step identifiers and dependenciesRule: scaled_fraction_zero_guard · Uses: G001, S005 S007 · calculationBefore scaled numerator1 := mul(5, -1) After scaled numerator1 = -5 Conditions: none Multiply the original numerator by this integer scale,retaining the exact integer product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore scaled denominator1 := mul(-3, -1) After scaled denominator1 = 3 Conditions: none Multiply the original denominator by the same scale,without prior reduction or deletion of zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore scaled rational value1 := div(-5, 3) After scaled rational value1 = -5/3 Conditions: none Reduce to a rational value only when the new denominator is nonzero. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore scale=-1;raw scaled pair=[-5, 3] After legal=Yes;normalized value=-5/3 Conditions: none A nonzero scale cancels and preserves value;zero scaling produces0/0,which is invalid even with zero numerator and cannot be reduced back to the original value. Step identifiers and dependenciesRule: scaled_fraction_zero_guard · Uses: G001, S009 ## Reverse checks - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute scaled rational value1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute scaled denominator1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute scaled numerator1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute scaled rational value0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute scaled denominator0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute scaled numerator0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute original fraction value 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, [[10, -6], [-5, 3]], [Yes, Yes], [-5/3, -5/3]) 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: 897f1cf0342bd660b7df6f6f4be4f771df31d6e2d8d0eb685d72b8661e9d7d7f - candidate record SHA-256: 627b7759896ff4a3e5f42ba79115731f1857744c1c11d388f3ba89e269a5b285 - 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-0093.json)