# Question 32120: Equivalence of Two Fractions [Return to catalogue page 161](https://ocme.evemisslab.com/en/questions/page/161/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260930-r01-fraction-foundations/#q-20260930-r01-fraction-equivalence-0020) 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. R0063-32120 · q-20260930-r01-fraction-equivalence-0020 ## Original question and givens The valid fractions are 7/(-6) and 36/(-30), with both denominators nonzero. Do they represent the same rational number? - G001 — The valid fractions are 7/(-6) and 36/(-30), with both denominators nonzero. Do they represent the same rational number? ## 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 source roles = [qf-fraction-equivalence, equivalence, no-required-domain, []] After derived = [[exact_equivalence_YesNo_STRING]] Conditions: none BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds. Step identifiers and dependenciesRule: R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle · Uses: G001 S002 · impliesBefore source roles = [7, -6, [36, -30]] After derived = [Yes] Conditions: none Validate EVERY raw SAFE integer component,including the numerator at q0/0/0 eligibility.A validity task may legally return No;actual arithmetic requires ALL used denominators nonzero. Step identifiers and dependenciesRule: Fraction_ALL_RAW_SAFE_integer_components_even_zero_den_validity · Uses: G001, S001 S003 · calculationBefore Fraction ORIGINAL p times s := mul(7, -30) After Fraction ORIGINAL p times s = -210 Conditions: none Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Fraction ORIGINAL r times q := mul(36, -6) After Fraction ORIGINAL r times q = -216 Conditions: none Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Fraction ORIGINAL cross product difference := sub(-210, -216) After Fraction ORIGINAL cross product difference = 6 Conditions: none Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [7, -6, 36, -30] After derived = [-210, -216, 6, No] Conditions: none With BOTH denominators nonzero,p/q=r/s IFF ps=rq.Negative denominators are legal.Compare complete exact products,not appearance or decimal approximation. Step identifiers and dependenciesRule: Fraction_EQUIVALENCE_nonzero_dens_EXACT_crossproducts_not_appearance · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S005 · premise_and_result_audit — Recompute Fraction ORIGINAL cross product difference with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Fraction ORIGINAL r times q with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Fraction ORIGINAL p times s with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S001 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. ## Result 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: 4c5658635067a942c3ac3eed6d3fca1557ec3570f70479b1b50a9a7f80f6fecc - candidate record SHA-256: 74093dce20e67e110de7d2dbbd674c6fb7d51f14f954fc251e06d58112d27f93 - semantic review receipt SHA-256: acf03b776f4855e3869810bcbecf1116e3a2b11dfb1e2a5dfbcf8135ad24a7d0 - question-first witness SHA-256: eadbbac3220988b52754f32450fb536d627217306d1ce28a49134f56c21bfa88 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260930-r01-fraction-equivalence-0020.json)