# Question 7048: Forward Ratio and Independent Reverse 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-0023)

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.

R0013-07048 · q-20261011-advance-20261015-r02-ratio-proportion-0023



## 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. a=21, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse 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. a=21, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse 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

forward ratio a/b := div(21, 3)

After

forward ratio a/b = 7

Conditions: none

The forward denominator b is nonzero,so compute the forward ratio.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

reverse ratio b/a := div(3, 21)

After

reverse ratio b/a = 1/7

Conditions: none

The reverse denominator is a and must separately be nonzero.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

forward times reverse := mul(7, 1/7)

After

forward times reverse = 1

Conditions: none

Multiply to audit reciprocity only when both directions are defined.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

a=21;b=3;forward=7/1

After

reverse defined=Yes;reverse=1/7

Conditions: none

When a=0,the valid forward ratio may be0,but this does not permit b/0 in reverse.Nor may the undefined product be assigned0.

Step identifiers and dependenciesRule: reverse_ratio_independent_denominator · Uses: G001, S004









## Reverse checks


- 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 forward times reverse with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute reverse ratio b/a with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute forward ratio a/b 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

(7/1, Yes, 1/7, 1/1)

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: 2bdac1ae22fa1fd346ce7dd4edf20a57092dc011c2035de4a95c59068688a17b
- candidate record SHA-256: e535ef4971cec2ecef4bfeb7bfb237626b427a20a5ed8ce57375e09ee7dcd69e
- 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-0023.json)