Question 2106: Synthetic Fraction-Equivalence Response with Nonzero Denominators

Original question and givens

Use a synthetic response record and the explicitly declared mathematical rule. A match code reports only rule agreement, not the ability, personality, learning difficulty, cause of error or learning outcome of a real student. Fractions p/q and r/s have p=-3, q=1≠0, r=-18, s=6≠0; Boolean response=true. Give (p,q,r,s,p*s,r*q,equivalent?,response,match?,rule code). Preserve the signed original numerator/denominator values in these fields.

• G001 — Use a synthetic response record and the explicitly declared mathematical rule. A match code reports only rule agreement, not the ability, personality, learning difficulty, cause of error or learning outcome of a real student. Fractions p/q and r/s have p=-3, q=1≠0, r=-18, s=6≠0; Boolean response=true. Give (p,q,r,s,p*s,r*q,equivalent?,response,match?,rule code). Preserve the signed original numerator/denominator values in these fields.

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-diagnostic-mathematical-assessment

Step-by-step derivation

synthetic response record and a declared mathematical rule

check rule agreement,not a learner diagnosis

denominators=1,6, both nonzero

a/b=c/d iff a·d=c·b

left cross-product := mul(-3, 6)

left cross-product = -18

right cross-product := mul(-18, 1)

right cross-product = -18

cross-products=-18,-18; equivalent=Yes

response=Yes; match=Yes

rule agreement=Yes

record code=matches_rule_expected

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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute right cross-product with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute left cross-product with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(-3, 1, -18, 6, -18, -18, Yes, Yes, Yes, matches_rule_expected)

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.