第 32141 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 3/(-2)+(-8)/(-1)，寫成最簡正分母 p/q。

• G001 — 對整數構成的合法分數，精確計算 3/(-2)+(-8)/(-1)，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 = [3, -2, [-8, -1]]

derived = [是]

Fraction ORIGINAL p times s := mul(3, -1)

Fraction ORIGINAL p times s = -3

Fraction ORIGINAL r times q := mul(-8, -2)

Fraction ORIGINAL r times q = 16

Fraction addition ORIGINAL cross numerator sum := sum(-3, 16)

Fraction addition ORIGINAL cross numerator sum = 13

Fraction addition ORIGINAL denominator product := mul(-2, -1)

Fraction addition ORIGINAL denominator product = 2

source roles = [3, -2, -8, -1]

derived = [13, 2]

source roles = [Fraction addition normalization, 13, 2]

derived = [13, 2]

Fraction addition normalization gcd step0 ORIGINAL quotient times divisor := mul(6, 2)

Fraction addition normalization gcd step0 ORIGINAL quotient times divisor = 12

Fraction addition normalization gcd step0 ORIGINAL dividend minus product := sub(13, 12)

Fraction addition normalization gcd step0 ORIGINAL dividend minus product = 1

source roles = [Fraction addition normalization, 0, 13, 2, 6]

derived = [12, 1]

Fraction addition normalization gcd step1 ORIGINAL quotient times divisor := mul(2, 1)

Fraction addition normalization gcd step1 ORIGINAL quotient times divisor = 2

Fraction addition normalization gcd step1 ORIGINAL dividend minus product := sub(2, 2)

Fraction addition normalization gcd step1 ORIGINAL dividend minus product = 0

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

derived = [2, 0]

Fraction addition normalization ORIGINAL numerator over positive gcd := div(13, 1)

Fraction addition normalization ORIGINAL numerator over positive gcd = 13

Fraction addition normalization ORIGINAL denominator over positive gcd := div(2, 1)

Fraction addition normalization ORIGINAL denominator over positive gcd = 2

Fraction addition normalization reverse reduced numerator times gcd := mul(13, 1)

Fraction addition normalization reverse reduced numerator times gcd = 13

Fraction addition normalization reverse reduced denominator times gcd := mul(2, 1)

Fraction addition normalization reverse reduced denominator times gcd = 2

source roles = [Fraction addition normalization, 13, 2, 1]

derived = [13, 2, 13, 2, 13/2]

Fraction addition reverse ORIGINAL fraction sum := sum(-3/2, 8)

Fraction addition reverse ORIGINAL fraction sum = 13/2

source roles = [-3/2, 8/1]

derived = [13/2, 13/2]

逐步反向核對

• 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 reverse ORIGINAL fraction sum，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 reverse reduced denominator times 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 reverse reduced numerator times gcd，不以小數近似替代。
• S016 · 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。
• S015 · 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。
• 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

13/2

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。