Question 32164: Fraction Reduction and Normalization

Original question and givens

Reduce the valid integer fraction 3/(-21) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

• G001 — Reduce the valid integer fraction 3/(-21) to lowest terms with a positive denominator. Write the answer as p/q, including integers and zero.

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

source roles = [qf-fraction-reduction, reduction, no-required-domain, []]

derived = [[reduced_fraction_RETAIN1_STRING]]

source roles = [3, -21, []]

derived = [Yes]

Fraction reduction ORIGINAL negate numerator for denominator sign := sub(0, 3)

Fraction reduction ORIGINAL negate numerator for denominator sign = -3

Fraction reduction ORIGINAL negate denominator := sub(0, -21)

Fraction reduction ORIGINAL negate denominator = 21

source roles = [Fraction reduction, 3, -21]

derived = [-3, 21]

Fraction reduction gcd step0 ORIGINAL quotient times divisor := mul(0, 21)

Fraction reduction gcd step0 ORIGINAL quotient times divisor = 0

Fraction reduction gcd step0 ORIGINAL dividend minus product := sub(3, 0)

Fraction reduction gcd step0 ORIGINAL dividend minus product = 3

source roles = [Fraction reduction, 0, 3, 21, 0]

derived = [0, 3]

Fraction reduction gcd step1 ORIGINAL quotient times divisor := mul(7, 3)

Fraction reduction gcd step1 ORIGINAL quotient times divisor = 21

Fraction reduction gcd step1 ORIGINAL dividend minus product := sub(21, 21)

Fraction reduction gcd step1 ORIGINAL dividend minus product = 0

source roles = [Fraction reduction, 1, 21, 3, 7]

derived = [21, 0]

Fraction reduction ORIGINAL numerator over positive gcd := div(-3, 3)

Fraction reduction ORIGINAL numerator over positive gcd = -1

Fraction reduction ORIGINAL denominator over positive gcd := div(21, 3)

Fraction reduction ORIGINAL denominator over positive gcd = 7

Fraction reduction reverse reduced numerator times gcd := mul(-1, 3)

Fraction reduction reverse reduced numerator times gcd = -3

Fraction reduction reverse reduced denominator times gcd := mul(7, 3)

Fraction reduction reverse reduced denominator times gcd = 21

source roles = [Fraction reduction, -3, 21, 3]

derived = [-1, 7, -3, 21, -1/7]

Reverse checks

• S016 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S015 · premise_and_result_audit — Recompute Fraction reduction reverse reduced denominator times gcd with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Fraction reduction reverse reduced numerator times gcd with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Fraction reduction ORIGINAL denominator over positive gcd with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Fraction reduction ORIGINAL numerator over positive gcd with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S010 · premise_and_result_audit — Recompute Fraction reduction gcd step1 ORIGINAL dividend minus product with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Fraction reduction gcd step1 ORIGINAL quotient times divisor with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S007 · premise_and_result_audit — Recompute Fraction reduction gcd step0 ORIGINAL dividend minus product with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Fraction reduction gcd step0 ORIGINAL quotient times divisor with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S004 · premise_and_result_audit — Recompute Fraction reduction ORIGINAL negate denominator with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Fraction reduction ORIGINAL negate numerator for denominator sign with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S001 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

-1/7

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.