# Question 7073: Candidate Proportion Guards and Cross-Product Defect [Return to catalogue page 36](https://ocme.evemisslab.com/en/questions/page/36/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261011-advance-20261015-r02-ratio-proportion-foundations/#q-20261011-advance-20261015-r02-ratio-proportion-0098) 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. R0013-07073 · q-20261011-advance-20261015-r02-ratio-proportion-0098 ## Original question and givens Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=10, b=3, d=2, x=23/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?). - G001 — Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=10, b=3, d=2, x=23/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?). ## 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-ratio-proportion](https://ocme.evemisslab.com/en/mko/mko-ratio-proportion/) ## Step-by-step derivation S001 · impliesBefore exact rational numerical ratios;each denominator needs its own guard After cross-multiplied equality does not legalize a zero ratio denominator Conditions: none A numerical ratio is not an automatic causal or geometric-similarity proof.Cross residual0 does not legalize a zero denominator,and the reverse ratio needs its own denominator guard. Step identifiers and dependenciesRule: rational_ratio_definition_scope · Uses: G001 S002 · calculationBefore cross product ad := mul(10, 2) After cross product ad = 20 Conditions: none Compute the cross term ad;this polynomial term exists even if the original ratio is illegal. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore cross product b times candidate := mul(3, 23/2) After cross product b times candidate = 69/2 Conditions: none Use the original candidate x in the second term,not a substituted solved x*. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore ad minus b times candidate := sub(20, 69/2) After ad minus b times candidate = -29/2 Conditions: none Keep the signed cross residual separate from eligibility of both original denominators. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore legal missing component x star := div(20, 3) After legal missing component x star = 20/3 Conditions: none Only when both original denominators are legal,solve for x* by ad/b. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore b=3;d=2;candidate=23/2;cross residual=-29/2 After original ratios legal=Yes;x*=20/3;candidate valid=No Conditions: none If b or d is0,the original proportion is illegal even when the cross residual is0.Solving for x* and auditing the supplied candidate are distinct tasks. Step identifiers and dependenciesRule: missing_ratio_component_original_guard · 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 — Recompute legal missing component x star with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute ad minus b times candidate with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute cross product b times candidate with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute cross product ad 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 (Yes, 20/3, -29/2, 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: 54576090656f286d5c7f75cf1ef1d5040b9fd5d5908426c3037847ed6f2ba8ee - candidate record SHA-256: 95024a07ea609bd730a20f6e77a2b09049aa0f25abe102c98cf48968808b55f4 - semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59 - question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261011-advance-20261015-r02-ratio-proportion-0098.json)