Question 6991: Two Raw Scaled Fraction Representations

Original question and givens

Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=3, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.

• G001 — Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=3, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.

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-fraction

Step-by-step derivation

original integer numerator/denominator representations;original denominators nonzero

representation and rational value are distinct data

original fraction value := div(3, -3)

original fraction value = -1

scaled numerator0 := mul(3, 2)

scaled numerator0 = 6

scaled denominator0 := mul(-3, 2)

scaled denominator0 = -6

scaled rational value0 := div(6, -6)

scaled rational value0 = -1

scale=2;raw scaled pair=[6, -6]

legal=Yes;normalized value=-1/1

scaled numerator1 := mul(3, -1)

scaled numerator1 = -3

scaled denominator1 := mul(-3, -1)

scaled denominator1 = 3

scaled rational value1 := div(-3, 3)

scaled rational value1 = -1

scale=-1;raw scaled pair=[-3, 3]

legal=Yes;normalized value=-1/1

Reverse checks

• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Recompute scaled rational value1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute scaled denominator1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute scaled numerator1 with exact rational arithmetic, not rounded decimals.
• 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 scaled rational value0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute scaled denominator0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute scaled numerator0 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute original fraction value 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

(-1/1, [[6, -6], [-3, 3]], [Yes, Yes], [-1/1, -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.