Question 34921: Exact Distance Between Rational Points

Original question and givens

Two rational real points are 10/1 and -8/7. Find the exact distance |10/1−-8/7| as a lowest-terms fraction.

• G001 — Two rational real points are 10/1 and -8/7. Find the exact distance |10/1−-8/7| as a lowest-terms fraction.

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-real-number

Step-by-step derivation

source roles = [qf-real-rational-point-distance, [left, operation, right]]

derived = [exact_rational_distance, no-invented-domain]

Real ORIGINAL left point ORIGINAL exact numerator over nonzero denominator := div(10, 1)

Real ORIGINAL left point ORIGINAL exact numerator over nonzero denominator = 10

source roles = [Real ORIGINAL left point, [10, 1]]

derived = [[10, 1], 10/1]

Real ORIGINAL right point ORIGINAL exact numerator over nonzero denominator := div(-8, 7)

Real ORIGINAL right point ORIGINAL exact numerator over nonzero denominator = -8/7

source roles = [Real ORIGINAL right point, [-8, 7]]

derived = [[-8, 7], -8/7]

Real ORIGINAL normalized left numerator times right positive denominator := mul(10, 7)

Real ORIGINAL normalized left numerator times right positive denominator = 70

Real ORIGINAL normalized right numerator times left positive denominator := mul(-8, 1)

Real ORIGINAL normalized right numerator times left positive denominator = -8

Real distance ORIGINAL signed crossproduct difference := sub(70, -8)

Real distance ORIGINAL signed crossproduct difference = 78

Real distance ORIGINAL positive denominator product := mul(1, 7)

Real distance ORIGINAL positive denominator product = 7

source roles = [10/1, -8/7, 78, 7]

derived = [78, 7]

Real exact distance derived fraction ORIGINAL exact numerator over nonzero denominator := div(78, 7)

Real exact distance derived fraction ORIGINAL exact numerator over nonzero denominator = 78/7

source roles = [Real exact distance derived fraction, [78, 7]]

derived = [[78, 7], 78/7]

Reverse checks

• S012 · 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.
• S011 · premise_and_result_audit — Recompute Real exact distance derived fraction ORIGINAL exact numerator over nonzero denominator with exact rational arithmetic, not rounded decimals.
• S010 · 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.
• S009 · premise_and_result_audit — Recompute Real distance ORIGINAL positive denominator product with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Real distance ORIGINAL signed crossproduct difference with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Real ORIGINAL normalized right numerator times left positive denominator with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Real ORIGINAL normalized left numerator times right positive denominator with exact rational arithmetic, not rounded decimals.
• S005 · 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.
• S004 · premise_and_result_audit — Recompute Real ORIGINAL right point ORIGINAL exact numerator over nonzero denominator with exact rational arithmetic, not rounded decimals.
• S003 · 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.
• S002 · premise_and_result_audit — Recompute Real ORIGINAL left point ORIGINAL exact numerator over nonzero denominator 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

78/7

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.