Question 1655: Relative Error with an Explicit Nonzero Reference

Original question and givens

Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=11/5, reference=5/1 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.

• G001 — Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=11/5, reference=5/1 is required to be nonzero. Give (approximation, reference, absolute error, absolute error divided by absolute reference). Do not convert the ratio to a percentage.

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-numerical-approximation-error

Step-by-step derivation

reference=5

all values are exact rationals

approximation−reference := sub(11/5, 5)

approximation−reference = -14/5

absolute error := mul(-1, -14/5)

absolute error = 14/5

signed error=-14/5

absolute error=14/5≥0

reference=5≠0

relative denominator=|reference|=5>0

relative error := div(14/5, 5)

relative error = 14/25

relative error=14/25

ratio may exceed1; it is not a probability

Reverse checks

• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute relative error with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Recompute absolute error with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute approximation−reference 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

(11/5, 5, 14/5, 14/25)

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.