第 32144 題：分數精確加法

原題與已知條件

對整數構成的合法分數，精確計算 6/(5)+(8)/(-4)，寫成最簡正分母 p/q。

• G001 — 對整數構成的合法分數，精確計算 6/(5)+(8)/(-4)，寫成最簡正分母 p/q。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-fraction

逐步推導

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

derived = [[reduced_fraction_RETAIN1_STRING]]

source roles = [6, 5, [8, -4]]

derived = [是]

Fraction ORIGINAL p times s := mul(6, -4)

Fraction ORIGINAL p times s = -24

Fraction ORIGINAL r times q := mul(8, 5)

Fraction ORIGINAL r times q = 40

Fraction addition ORIGINAL cross numerator sum := sum(-24, 40)

Fraction addition ORIGINAL cross numerator sum = 16

Fraction addition ORIGINAL denominator product := mul(5, -4)

Fraction addition ORIGINAL denominator product = -20

source roles = [6, 5, 8, -4]

derived = [16, -20]

Fraction addition normalization ORIGINAL negate numerator for denominator sign := sub(0, 16)

Fraction addition normalization ORIGINAL negate numerator for denominator sign = -16

Fraction addition normalization ORIGINAL negate denominator := sub(0, -20)

Fraction addition normalization ORIGINAL negate denominator = 20

source roles = [Fraction addition normalization, 16, -20]

derived = [-16, 20]

Fraction addition normalization gcd step0 ORIGINAL quotient times divisor := mul(0, 20)

Fraction addition normalization gcd step0 ORIGINAL quotient times divisor = 0

Fraction addition normalization gcd step0 ORIGINAL dividend minus product := sub(16, 0)

Fraction addition normalization gcd step0 ORIGINAL dividend minus product = 16

source roles = [Fraction addition normalization, 0, 16, 20, 0]

derived = [0, 16]

Fraction addition normalization gcd step1 ORIGINAL quotient times divisor := mul(1, 16)

Fraction addition normalization gcd step1 ORIGINAL quotient times divisor = 16

Fraction addition normalization gcd step1 ORIGINAL dividend minus product := sub(20, 16)

Fraction addition normalization gcd step1 ORIGINAL dividend minus product = 4

source roles = [Fraction addition normalization, 1, 20, 16, 1]

derived = [16, 4]

Fraction addition normalization gcd step2 ORIGINAL quotient times divisor := mul(4, 4)

Fraction addition normalization gcd step2 ORIGINAL quotient times divisor = 16

Fraction addition normalization gcd step2 ORIGINAL dividend minus product := sub(16, 16)

Fraction addition normalization gcd step2 ORIGINAL dividend minus product = 0

source roles = [Fraction addition normalization, 2, 16, 4, 4]

derived = [16, 0]

Fraction addition normalization ORIGINAL numerator over positive gcd := div(-16, 4)

Fraction addition normalization ORIGINAL numerator over positive gcd = -4

Fraction addition normalization ORIGINAL denominator over positive gcd := div(20, 4)

Fraction addition normalization ORIGINAL denominator over positive gcd = 5

Fraction addition normalization reverse reduced numerator times gcd := mul(-4, 4)

Fraction addition normalization reverse reduced numerator times gcd = -16

Fraction addition normalization reverse reduced denominator times gcd := mul(5, 4)

Fraction addition normalization reverse reduced denominator times gcd = 20

source roles = [Fraction addition normalization, -16, 20, 4]

derived = [-4, 5, -16, 20, -4/5]

Fraction addition reverse ORIGINAL fraction sum := sum(6/5, -2)

Fraction addition reverse ORIGINAL fraction sum = -4/5

source roles = [6/5, -2/1]

derived = [-4/5, -4/5]

逐步反向核對

• S026 · premise_and_result_audit — 依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。
• S025 · premise_and_result_audit — 以精確有理數重算 Fraction addition reverse ORIGINAL fraction sum，不以小數近似替代。
• S024 · premise_and_result_audit — 依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。
• S023 · premise_and_result_audit — 以精確有理數重算 Fraction addition normalization reverse reduced denominator times gcd，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Fraction addition normalization reverse reduced numerator times gcd，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Fraction addition normalization ORIGINAL denominator over positive gcd，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Fraction addition normalization ORIGINAL numerator over positive gcd，不以小數近似替代。
• S019 · premise_and_result_audit — 依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。
• S018 · premise_and_result_audit — 以精確有理數重算 Fraction addition normalization gcd step2 ORIGINAL dividend minus product，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Fraction addition normalization gcd step2 ORIGINAL quotient times divisor，不以小數近似替代。
• S016 · premise_and_result_audit — 依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。
• S015 · premise_and_result_audit — 以精確有理數重算 Fraction addition normalization gcd step1 ORIGINAL dividend minus product，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Fraction addition normalization gcd step1 ORIGINAL quotient times divisor，不以小數近似替代。
• S013 · premise_and_result_audit — 依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。
• S012 · premise_and_result_audit — 以精確有理數重算 Fraction addition normalization gcd step0 ORIGINAL dividend minus product，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Fraction addition normalization gcd step0 ORIGINAL quotient times divisor，不以小數近似替代。
• S010 · premise_and_result_audit — 依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。
• S009 · premise_and_result_audit — 以精確有理數重算 Fraction addition normalization ORIGINAL negate denominator，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Fraction addition normalization ORIGINAL negate numerator for denominator sign，不以小數近似替代。
• S007 · premise_and_result_audit — 依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。
• S006 · premise_and_result_audit — 以精確有理數重算 Fraction addition ORIGINAL denominator product，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Fraction addition ORIGINAL cross numerator sum，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Fraction ORIGINAL r times q，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Fraction ORIGINAL p times s，不以小數近似替代。
• S002 · premise_and_result_audit — 依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。
• S001 · premise_and_result_audit — 依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。

結論

-4/5

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。