# Question 7087: Same-Unit Weighted Partition and Actual Ratio Guard [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-0062) 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. R0013-07087 · q-20261011-advance-20261015-r02-ratio-proportion-0062 ## 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. Positive weights a=12, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined). - G001 — Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=12, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined). ## 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 positive weight sum := sum(12, 2) After positive weight sum = 14 Conditions: none The sum of two positive weights is positive and is a legal allocation denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore total times first weight := mul(5/2, 12) After total times first weight = 30 Conditions: none Multiply the total by the first weight. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore first proportional part x := div(30, 14) After first proportional part x = 15/7 Conditions: none Divide by the positive weight sum to obtain the first part. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore second part from remaining total := sub(5/2, 15/7) After second part from remaining total = 5/14 Conditions: none Subtract the first part from the same-unit total to obtain the second. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore sum of allocated parts := sum(15/7, 5/14) After sum of allocated parts = 5/2 Conditions: none Add the two parts to audit the original total. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore allocation sum residual := sub(5/2, 5/2) After allocation sum residual = 0 Conditions: none The restored sum minus the original total must be0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore allocation b times x := mul(2, 15/7) After allocation b times x = 30/7 Conditions: none Compute b·x as the first cross term. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore allocation a times y := mul(12, 5/14) After allocation a times y = 30/7 Conditions: none Compute a·y as the second cross term. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore allocation proportional residual := sub(30/7, 30/7) After allocation proportional residual = 0 Conditions: none The cross residual b·x−a·y must be0,but does not replace the actual ratio-denominator check. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore actual allocated ratio x/y := div(15/7, 5/14) After actual allocated ratio x/y = 6 Conditions: none The actual second part must be nonzero to give x:y a numerical value. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore T=5/2;parts=(15/7,5/14);residuals=(0,0) After actual ratio legal=Yes;x/y=6/1 Conditions: none A zero total validly allocates0 and0 with both residuals0,but0:0 has no numerical ratio.The positive declared weights do not waive the actual denominator check. Step identifiers and dependenciesRule: zero_total_not_valid_zero_ratio · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute actual allocated ratio x/y with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute allocation proportional residual with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute allocation a times y with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute allocation b times x with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute allocation sum residual with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute sum of allocated parts with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute second part from remaining total with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute first proportional part x with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute total times first weight with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute positive weight sum 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 (15/7, 5/14, 0/1, 0/1, Yes, 6/1) 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: 24736bbb2e5982db86a5b938e9b143431bc1b0ccdd064464428b82048b1a4d91 - candidate record SHA-256: 1fe4cfd1ae8ae46453eb9e65cb24fdfbdd73769e93197c2aa2619a436bc7194f - 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-0062.json)