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R0056-28671 · q-20261005-advance-20261006-r03-rate-of-change-0021
For the same specified function, the permitted endpoints are (7, 31) and (14, 52), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.
G001 — For the same specified function, the permitted endpoints are (7, 31) and (14, 52), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.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.
Before
source roles = [qf-rate-endpoint-quotient, endpoint_quotient]
After
derived = [exact_oriented_difference_quotient]
Conditions: none
Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions.
Rule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001
Before
Rate ORIGINAL x1 minus x0 := sub(14, 7)
After
Rate ORIGINAL x1 minus x0 = 7
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
Rate ORIGINAL y1 minus y0 := sub(52, 31)
After
Rate ORIGINAL y1 minus y0 = 21
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Rate ORIGINAL dy over dx := div(21, 7)
After
Rate ORIGINAL dy over dx = 3
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
source roles = [7, 14, 31, 52, 7, 21]
After
derived = [3/1]
Conditions: none
Keep the SAME endpoint orientation:(y1-y0)/(x1-x0).Both differences may reverse together,not just one.The result may be negative or0;normalize a positive denominator and retain/1.This finite endpoint rate is not a general derivative certificate.
Rule: Rate_oriented_quotient_reverse_BOTH_differences · Uses: G001, S004
S005 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.S004 · premise_and_result_audit — Recompute Rate ORIGINAL dy over dx with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Rate ORIGINAL y1 minus y0 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Rate ORIGINAL x1 minus x0 with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.3/1
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 7111acf66f94e7e37c2f10a04fe9b2815210357a7c462088052eb9ae319b49fecandidate record SHA-256: a6050daf56c2a8b22c8c46993c446be57430e39b0248a31b3975e39394cbcdacsemantic review receipt SHA-256: 8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8question-first witness SHA-256: 3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759fLean: NOT_RUN · human review: NOT_PERFORMED