Question 1636: Inclusive Error Interval and a Candidate Boundary

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. reference=2/3, tolerance=5/5≥0, candidate=21/1. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).

• 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. reference=2/3, tolerance=5/5≥0, candidate=21/1. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).

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=2/3

all values are exact rationals

reference−tolerance := sub(2/3, 1)

reference−tolerance = -1/3

reference+tolerance := sum(2/3, 1)

reference+tolerance = 5/3

|candidate−reference|≤tolerance

reference−tolerance≤candidate≤reference+tolerance

candidate−reference := sub(21, 2/3)

candidate−reference = 61/3

candidate=21; |deviation|=61/3

inside closed interval: No

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 — Recompute candidate−reference with exact rational arithmetic, not rounded decimals.
• 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 reference+tolerance with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute reference−tolerance 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

(2/3, 1, -1/3, 5/3, 21, No)

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.