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Question 7075: Candidate Proportion Guards and Cross-Product Defect

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0013-07075 · q-20261011-advance-20261015-r02-ratio-proportion-0100

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=12, b=3, d=2, x=25/2. 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=12, b=3, d=2, x=25/2. 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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: rational_ratio_definition_scope · Uses: G001

S002 · calculation

Before

cross product ad := mul(12, 2)

After

cross product ad = 24

Conditions: none

Compute the cross term ad;this polynomial term exists even if the original ratio is illegal.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

cross product b times candidate := mul(3, 25/2)

After

cross product b times candidate = 75/2

Conditions: none

Use the original candidate x in the second term,not a substituted solved x*.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

ad minus b times candidate := sub(24, 75/2)

After

ad minus b times candidate = -27/2

Conditions: none

Keep the signed cross residual separate from eligibility of both original denominators.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

legal missing component x star := div(24, 3)

After

legal missing component x star = 8

Conditions: none

Only when both original denominators are legal,solve for x* by ad/b.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

b=3;d=2;candidate=25/2;cross residual=-27/2

After

original ratios legal=Yes;x*=8/1;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 dependencies

Rule: 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, 8/1, -27/2, 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: 4bb846364bb2316ad0130dd6d63777e7d44aad3ae4a63040f75c155a0c9254ea
  • candidate record SHA-256: f301043be79b8db226851962d336463142f0aa5261618be143e61482d5a6f04f
  • semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59
  • question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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