第 32146 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/(7)+(2)/(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 8/(7)+(2)/(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 = [8, 7, [2, 1]]

derived = [是]

Fraction ORIGINAL p times s := mul(8, 1)

Fraction ORIGINAL p times s = 8

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

Fraction ORIGINAL r times q = 14

Fraction addition ORIGINAL cross numerator sum := sum(8, 14)

Fraction addition ORIGINAL cross numerator sum = 22

Fraction addition ORIGINAL denominator product := mul(7, 1)

Fraction addition ORIGINAL denominator product = 7

source roles = [8, 7, 2, 1]

derived = [22, 7]

source roles = [Fraction addition normalization, 22, 7]

derived = [22, 7]

Fraction addition normalization gcd step0 ORIGINAL quotient times divisor := mul(3, 7)

Fraction addition normalization gcd step0 ORIGINAL quotient times divisor = 21

Fraction addition normalization gcd step0 ORIGINAL dividend minus product := sub(22, 21)

Fraction addition normalization gcd step0 ORIGINAL dividend minus product = 1

source roles = [Fraction addition normalization, 0, 22, 7, 3]

derived = [21, 1]

Fraction addition normalization gcd step1 ORIGINAL quotient times divisor := mul(7, 1)

Fraction addition normalization gcd step1 ORIGINAL quotient times divisor = 7

Fraction addition normalization gcd step1 ORIGINAL dividend minus product := sub(7, 7)

Fraction addition normalization gcd step1 ORIGINAL dividend minus product = 0

source roles = [Fraction addition normalization, 1, 7, 1, 7]

derived = [7, 0]

Fraction addition normalization ORIGINAL numerator over positive gcd := div(22, 1)

Fraction addition normalization ORIGINAL numerator over positive gcd = 22

Fraction addition normalization ORIGINAL denominator over positive gcd := div(7, 1)

Fraction addition normalization ORIGINAL denominator over positive gcd = 7

Fraction addition normalization reverse reduced numerator times gcd := mul(22, 1)

Fraction addition normalization reverse reduced numerator times gcd = 22

Fraction addition normalization reverse reduced denominator times gcd := mul(7, 1)

Fraction addition normalization reverse reduced denominator times gcd = 7

source roles = [Fraction addition normalization, 22, 7, 1]

derived = [22, 7, 22, 7, 22/7]

Fraction addition reverse ORIGINAL fraction sum := sum(8/7, 2)

Fraction addition reverse ORIGINAL fraction sum = 22/7

source roles = [8/7, 2/1]

derived = [22/7, 22/7]

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

22/7

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。