# Question 34794: Equivalent Fractions

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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.

R0068-34794 · q-20260920-rational-equivalent-fractions-0019



## Original question and givens

The denominators q=5 and s=10 are nonzero. Are 7/5 and 14/10 equivalent?


- G001 — The denominators q=5 and s=10 are nonzero. Are 7/5 and 14/10 equivalent?





## 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-rational-number](https://ocme.evemisslab.com/en/mko/mko-rational-number/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-rational-equivalent-fractions, [p, q, r, s]]

After

derived = [BASE-no-required-operation, no-invented-domain]

Conditions: none

Separate source banks:legacy RAW SAFE with unbounded derived BigInts versus BASE Number.isInteger without invented SAFE/million caps.Operations,domains and permissive/exact keys follow THIS bank.Keep ORIGINAL JSON intact.

Step identifiers and dependenciesRule: R68_BANK_specific_keys_operations_and_numeric_guards · Uses: G001





S002 · calculationBefore

Rational equivalence ORIGINAL p times s := mul(7, 10)

After

Rational equivalence ORIGINAL p times s = 70

Conditions: none

Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer.

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





S003 · calculationBefore

Rational equivalence ORIGINAL r times q := mul(14, 5)

After

Rational equivalence ORIGINAL r times q = 70

Conditions: none

Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer.

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





S004 · impliesBefore

source roles = [7, 5, 14, 10]

After

derived = [70, 70, Yes]

Conditions: none

FIRST qualify BOTH nonzero denominators,THEN compare p*s=r*q.Negative denominators are allowed for EQUATIONS,unlike order-direction issues.Do not compare decimal appearance or accept0=0 with an undefined ratio.

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









## Reverse checks


- S004 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.
- S003 · premise_and_result_audit — Recompute Rational equivalence ORIGINAL r times q with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute Rational equivalence ORIGINAL p times s with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.





## Result

Yes

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: d417d09c297b13adaf231221320e2eef2b6471f89e8ec963f788dd76769dd7b8
- candidate record SHA-256: c0426f7a1eb3f0299541cafb7566d7fa04c1637e472937fbda7c24899c398abe
- semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed
- question-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260920-rational-equivalent-fractions-0019.json)