第 32139 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 1/(-7)+(-2)/(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 1/(-7)+(-2)/(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 = [1, -7, [-2, 4]]

derived = [是]

Fraction ORIGINAL p times s := mul(1, 4)

Fraction ORIGINAL p times s = 4

Fraction ORIGINAL r times q := mul(-2, -7)

Fraction ORIGINAL r times q = 14

Fraction addition ORIGINAL cross numerator sum := sum(4, 14)

Fraction addition ORIGINAL cross numerator sum = 18

Fraction addition ORIGINAL denominator product := mul(-7, 4)

Fraction addition ORIGINAL denominator product = -28

source roles = [1, -7, -2, 4]

derived = [18, -28]

Fraction addition normalization ORIGINAL negate numerator for denominator sign := sub(0, 18)

Fraction addition normalization ORIGINAL negate numerator for denominator sign = -18

Fraction addition normalization ORIGINAL negate denominator := sub(0, -28)

Fraction addition normalization ORIGINAL negate denominator = 28

source roles = [Fraction addition normalization, 18, -28]

derived = [-18, 28]

Fraction addition normalization gcd step0 ORIGINAL quotient times divisor := mul(0, 28)

Fraction addition normalization gcd step0 ORIGINAL quotient times divisor = 0

Fraction addition normalization gcd step0 ORIGINAL dividend minus product := sub(18, 0)

Fraction addition normalization gcd step0 ORIGINAL dividend minus product = 18

source roles = [Fraction addition normalization, 0, 18, 28, 0]

derived = [0, 18]

Fraction addition normalization gcd step1 ORIGINAL quotient times divisor := mul(1, 18)

Fraction addition normalization gcd step1 ORIGINAL quotient times divisor = 18

Fraction addition normalization gcd step1 ORIGINAL dividend minus product := sub(28, 18)

Fraction addition normalization gcd step1 ORIGINAL dividend minus product = 10

source roles = [Fraction addition normalization, 1, 28, 18, 1]

derived = [18, 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(18, 10)

Fraction addition normalization gcd step2 ORIGINAL dividend minus product = 8

source roles = [Fraction addition normalization, 2, 18, 10, 1]

derived = [10, 8]

Fraction addition normalization gcd step3 ORIGINAL quotient times divisor := mul(1, 8)

Fraction addition normalization gcd step3 ORIGINAL quotient times divisor = 8

Fraction addition normalization gcd step3 ORIGINAL dividend minus product := sub(10, 8)

Fraction addition normalization gcd step3 ORIGINAL dividend minus product = 2

source roles = [Fraction addition normalization, 3, 10, 8, 1]

derived = [8, 2]

Fraction addition normalization gcd step4 ORIGINAL quotient times divisor := mul(4, 2)

Fraction addition normalization gcd step4 ORIGINAL quotient times divisor = 8

Fraction addition normalization gcd step4 ORIGINAL dividend minus product := sub(8, 8)

Fraction addition normalization gcd step4 ORIGINAL dividend minus product = 0

source roles = [Fraction addition normalization, 4, 8, 2, 4]

derived = [8, 0]

Fraction addition normalization ORIGINAL numerator over positive gcd := div(-18, 2)

Fraction addition normalization ORIGINAL numerator over positive gcd = -9

Fraction addition normalization ORIGINAL denominator over positive gcd := div(28, 2)

Fraction addition normalization ORIGINAL denominator over positive gcd = 14

Fraction addition normalization reverse reduced numerator times gcd := mul(-9, 2)

Fraction addition normalization reverse reduced numerator times gcd = -18

Fraction addition normalization reverse reduced denominator times gcd := mul(14, 2)

Fraction addition normalization reverse reduced denominator times gcd = 28

source roles = [Fraction addition normalization, -18, 28, 2]

derived = [-9, 14, -18, 28, -9/14]

Fraction addition reverse ORIGINAL fraction sum := sum(-1/7, -1/2)

Fraction addition reverse ORIGINAL fraction sum = -9/14

source roles = [-1/7, -1/2]

derived = [-9/14, -9/14]

逐步反向核對

• S032 · 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。
• S031 · 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。
• S030 · 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。
• S029 · 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。
• S028 · 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。
• S027 · 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。
• S026 · 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。
• 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 gcd step4 ORIGINAL dividend minus product，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Fraction addition normalization gcd step4 ORIGINAL quotient times divisor，不以小數近似替代。
• S022 · 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。
• S021 · 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。
• S020 · 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。
• 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

-9/14

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。