第 32145 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 7/(-6)+(5)/(5)，寫成最簡正分母 p/q。

• G001 — 對整數構成的合法分數，精確計算 7/(-6)+(5)/(5)，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 = [7, -6, [5, 5]]

derived = [是]

Fraction ORIGINAL p times s := mul(7, 5)

Fraction ORIGINAL p times s = 35

Fraction ORIGINAL r times q := mul(5, -6)

Fraction ORIGINAL r times q = -30

Fraction addition ORIGINAL cross numerator sum := sum(35, -30)

Fraction addition ORIGINAL cross numerator sum = 5

Fraction addition ORIGINAL denominator product := mul(-6, 5)

Fraction addition ORIGINAL denominator product = -30

source roles = [7, -6, 5, 5]

derived = [5, -30]

Fraction addition normalization ORIGINAL negate numerator for denominator sign := sub(0, 5)

Fraction addition normalization ORIGINAL negate numerator for denominator sign = -5

Fraction addition normalization ORIGINAL negate denominator := sub(0, -30)

Fraction addition normalization ORIGINAL negate denominator = 30

source roles = [Fraction addition normalization, 5, -30]

derived = [-5, 30]

Fraction addition normalization gcd step0 ORIGINAL quotient times divisor := mul(0, 30)

Fraction addition normalization gcd step0 ORIGINAL quotient times divisor = 0

Fraction addition normalization gcd step0 ORIGINAL dividend minus product := sub(5, 0)

Fraction addition normalization gcd step0 ORIGINAL dividend minus product = 5

source roles = [Fraction addition normalization, 0, 5, 30, 0]

derived = [0, 5]

Fraction addition normalization gcd step1 ORIGINAL quotient times divisor := mul(6, 5)

Fraction addition normalization gcd step1 ORIGINAL quotient times divisor = 30

Fraction addition normalization gcd step1 ORIGINAL dividend minus product := sub(30, 30)

Fraction addition normalization gcd step1 ORIGINAL dividend minus product = 0

source roles = [Fraction addition normalization, 1, 30, 5, 6]

derived = [30, 0]

Fraction addition normalization ORIGINAL numerator over positive gcd := div(-5, 5)

Fraction addition normalization ORIGINAL numerator over positive gcd = -1

Fraction addition normalization ORIGINAL denominator over positive gcd := div(30, 5)

Fraction addition normalization ORIGINAL denominator over positive gcd = 6

Fraction addition normalization reverse reduced numerator times gcd := mul(-1, 5)

Fraction addition normalization reverse reduced numerator times gcd = -5

Fraction addition normalization reverse reduced denominator times gcd := mul(6, 5)

Fraction addition normalization reverse reduced denominator times gcd = 30

source roles = [Fraction addition normalization, -5, 30, 5]

derived = [-1, 6, -5, 30, -1/6]

Fraction addition reverse ORIGINAL fraction sum := sum(-7/6, 1)

Fraction addition reverse ORIGINAL fraction sum = -1/6

source roles = [-7/6, 1/1]

derived = [-1/6, -1/6]

逐步反向核對

• S023 · 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。
• S022 · 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。
• S021 · 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。
• S020 · 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。
• S019 · 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。
• S018 · 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。
• S017 · 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。
• 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

-1/6

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。