第 32130 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 -8/(5)+(0)/(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 -8/(5)+(0)/(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 = [-8, 5, [0, 5]]

derived = [是]

Fraction ORIGINAL p times s := mul(-8, 5)

Fraction ORIGINAL p times s = -40

Fraction ORIGINAL r times q := mul(0, 5)

Fraction ORIGINAL r times q = 0

Fraction addition ORIGINAL cross numerator sum := sum(-40, 0)

Fraction addition ORIGINAL cross numerator sum = -40

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

Fraction addition ORIGINAL denominator product = 25

source roles = [-8, 5, 0, 5]

derived = [-40, 25]

source roles = [Fraction addition normalization, -40, 25]

derived = [-40, 25]

Fraction addition normalization gcd step0 ORIGINAL quotient times divisor := mul(1, 25)

Fraction addition normalization gcd step0 ORIGINAL quotient times divisor = 25

Fraction addition normalization gcd step0 ORIGINAL dividend minus product := sub(40, 25)

Fraction addition normalization gcd step0 ORIGINAL dividend minus product = 15

source roles = [Fraction addition normalization, 0, 40, 25, 1]

derived = [25, 15]

Fraction addition normalization gcd step1 ORIGINAL quotient times divisor := mul(1, 15)

Fraction addition normalization gcd step1 ORIGINAL quotient times divisor = 15

Fraction addition normalization gcd step1 ORIGINAL dividend minus product := sub(25, 15)

Fraction addition normalization gcd step1 ORIGINAL dividend minus product = 10

source roles = [Fraction addition normalization, 1, 25, 15, 1]

derived = [15, 10]

Fraction addition normalization gcd step2 ORIGINAL quotient times divisor := mul(1, 10)

Fraction addition normalization gcd step2 ORIGINAL quotient times divisor = 10

Fraction addition normalization gcd step2 ORIGINAL dividend minus product := sub(15, 10)

Fraction addition normalization gcd step2 ORIGINAL dividend minus product = 5

source roles = [Fraction addition normalization, 2, 15, 10, 1]

derived = [10, 5]

Fraction addition normalization gcd step3 ORIGINAL quotient times divisor := mul(2, 5)

Fraction addition normalization gcd step3 ORIGINAL quotient times divisor = 10

Fraction addition normalization gcd step3 ORIGINAL dividend minus product := sub(10, 10)

Fraction addition normalization gcd step3 ORIGINAL dividend minus product = 0

source roles = [Fraction addition normalization, 3, 10, 5, 2]

derived = [10, 0]

Fraction addition normalization ORIGINAL numerator over positive gcd := div(-40, 5)

Fraction addition normalization ORIGINAL numerator over positive gcd = -8

Fraction addition normalization ORIGINAL denominator over positive gcd := div(25, 5)

Fraction addition normalization ORIGINAL denominator over positive gcd = 5

Fraction addition normalization reverse reduced numerator times gcd := mul(-8, 5)

Fraction addition normalization reverse reduced numerator times gcd = -40

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

Fraction addition normalization reverse reduced denominator times gcd = 25

source roles = [Fraction addition normalization, -40, 25, 5]

derived = [-8, 5, -40, 25, -8/5]

Fraction addition reverse ORIGINAL fraction sum := sum(-8/5, 0)

Fraction addition reverse ORIGINAL fraction sum = -8/5

source roles = [-8/5, 0/1]

derived = [-8/5, -8/5]

逐步反向核對

• S027 · 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。
• S026 · 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。
• S025 · 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。
• S024 · 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。
• S023 · 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。
• S022 · 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。
• S021 · 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。
• S020 · 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。
• S019 · premise_and_result_audit — 以精確有理數重算 Fraction addition normalization gcd step3 ORIGINAL dividend minus product，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Fraction addition normalization gcd step3 ORIGINAL quotient times divisor，不以小數近似替代。
• S017 · 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。
• S016 · 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。
• S015 · 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。
• S014 · 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。
• S013 · 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。
• S012 · 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。
• S011 · 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。
• S010 · 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。
• S009 · 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。
• S008 · 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。
• 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

-8/5

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。