# Question 2109: Synthetic Fraction-Equivalence Response with Nonzero Denominators [Return to catalogue page 11](https://ocme.evemisslab.com/en/questions/page/11/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261021-r02-diagnostic-mathematical-assessment-foundations/#q-20261013-advance-20261021-r02-diagnostic-mathematical-assessment-0034) 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. R0003-02109 · q-20261013-advance-20261021-r02-diagnostic-mathematical-assessment-0034 ## Original question and givens Use a synthetic response record and the explicitly declared mathematical rule. A match code reports only rule agreement, not the ability, personality, learning difficulty, cause of error or learning outcome of a real student. Fractions p/q and r/s have p=0, q=4≠0, r=1, s=2≠0; Boolean response=true. Give (p,q,r,s,p*s,r*q,equivalent?,response,match?,rule code). Preserve the signed original numerator/denominator values in these fields. - G001 — Use a synthetic response record and the explicitly declared mathematical rule. A match code reports only rule agreement, not the ability, personality, learning difficulty, cause of error or learning outcome of a real student. Fractions p/q and r/s have p=0, q=4≠0, r=1, s=2≠0; Boolean response=true. Give (p,q,r,s,p*s,r*q,equivalent?,response,match?,rule code). Preserve the signed original numerator/denominator values in these fields. ## 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-diagnostic-mathematical-assessment](https://ocme.evemisslab.com/en/mko/mko-diagnostic-mathematical-assessment/) ## Step-by-step derivation S001 · impliesBefore synthetic response record and a declared mathematical rule After check rule agreement,not a learner diagnosis Conditions: none These are synthetic records;one response does not diagnose a real learner's ability,psychology or learning outcome. Step identifiers and dependenciesRule: synthetic_record_scope · Uses: G001 S002 · impliesBefore denominators=4,2, both nonzero After a/b=c/d iff a·d=c·b Conditions: none Multiply an equality by the same nonzero denominator product;negative denominators do not reverse equality. Step identifiers and dependenciesRule: fraction_equality_guard · Uses: G001, S001 S003 · calculationBefore left cross-product := mul(0, 2) After left cross-product = 0 Conditions: none Multiply the left numerator by the right denominator,retaining the original integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore right cross-product := mul(1, 4) After right cross-product = 4 Conditions: none Multiply the right numerator by the left denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore cross-products=0,4; equivalent=No After response=Yes; match=No Conditions: none Determine the correct equivalence truth first,then compare the Boolean response;False is a valid response,not missing data. Step identifiers and dependenciesRule: boolean_response_match · Uses: G001, S004 S006 · impliesBefore rule agreement=No After record code=differs_from_rule_expected Conditions: none The code records agreement with the stated rule,not a person-level score or general learning diagnosis. Step identifiers and dependenciesRule: diagnostic_rule_code · Uses: G001, S005 ## Reverse checks - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute right cross-product with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute left cross-product with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (0, 4, 1, 2, 0, 4, No, Yes, No, differs_from_rule_expected) 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: cf23b9ab149d341f20d5d0c90d73c0a9350f1c4311bb24f82e6f6d5f772eaaeb - candidate record SHA-256: a10f9ebc23fea02f33e764eb929350026f31ba43eb05b3e6c436e0bd16dfb510 - semantic review receipt SHA-256: 5dbc70ebd5f74b23384c93b63a2b1d6519e206c79f22ae16a6c2d7483cea783f - question-first witness SHA-256: cf121c8dd2fec13aa118b04b049ee39494f1d6d9ecdc13514cafb78c0fc021fa - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261021-r02-diagnostic-mathematical-assessment-0034.json)