{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0001","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae62ea5250f3f1eb2345ea20369b6b6ac1748c7cee17893311a348286eec5583","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133001,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"-1/2\",\"1/3\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=1/-2, b=1/3. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/6, -5/6)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-1/6, -5/6)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0002","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94f30e607cd609f9c743c257fe363220f9d0117317094b5dce7932f10db51f8a","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133002,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"0/1\",\"2/3\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=0/-5, b=-2/-3. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(2/3, -2/3)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (2/3, -2/3)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0003","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff336527a68978216cf1dd4b27a7839c63b0daae7fe467b616ca352c0a14d6dc","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133003,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"-10/3\",\"3/5\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=-10/3, b=3/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-41/15, -59/15)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-41/15, -59/15)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0004","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d424cd79858717f86e67c1372cddce42053be1f630aa6ccd72762927f50e762","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133004,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"-9/1\",\"1/1\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=-9/1, b=-4/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-8/1, -10/1)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-8/1, -10/1)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0005","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54fbc98e3c9ce4befebb77ea24254b2f23d5af8b6a67ccbbaf843eb8aa2d726d","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133005,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"-4/1\",\"1/1\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=-8/2, b=5/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-3/1, -5/1)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-3/1, -5/1)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0006","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a904776092de25322dcc68ec21fa427366877a2d92b27c4102f691cc610653e","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133006,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"-7/3\",\"3/2\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=-7/3, b=-6/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-5/6, -23/6)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-5/6, -23/6)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0007","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"474f65e4d4130f30f559a0c545bd7e7f8dc8d8a0abeb412cc8f04a8f5f61020d","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133007,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"-6/1\",\"7/5\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=-6/1, b=7/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-23/5, -37/5)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-23/5, -37/5)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0008","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b7edd8f20b1cada3e096f0e030ccb74b066f7df51a09f6a1a7cb5a70bd41311","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133008,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"-5/2\",\"2/1\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=-5/2, b=-8/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/2, -9/2)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-1/2, -9/2)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0009","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d632ef4ee6bd686ec991c6f6bd9830bc0e43bd28619a074ff02803164cc4b87a","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133009,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"-4/3\",\"9/5\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=-4/3, b=9/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(7/15, -47/15)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (7/15, -47/15)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0010","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a472878c743705fa8570881979f3ae642f16b351eea3772e5113b90b095a0eb0","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133010,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"-3/1\",\"5/2\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=-3/1, b=-10/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/2, -11/2)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-1/2, -11/2)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0011","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26cb319be735673b136eb6304c46273a99ff323feae993d8549d86d6e177dcb3","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133011,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"-1/1\",\"11/5\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=-2/2, b=11/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(6/5, -16/5)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (6/5, -16/5)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0012","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08f756d330f91683e516336d7573e1b7e3f59edbe2ed97e5fb5d61fc3ddc1e4f","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133012,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"-1/3\",\"3/1\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=-1/3, b=-12/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(8/3, -10/3)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (8/3, -10/3)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0013","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c890bc0506261afc93b283d5de149226f68ac6dd1cda8f1150e38e51c8b81552","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133013,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"0/1\",\"13/5\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=0/1, b=13/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(13/5, -13/5)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (13/5, -13/5)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0014","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2af4704300d9715b672daa3df81f73d3f4e0c35fbc6351eea40a10732386306","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133014,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"1/2\",\"7/2\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-14/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(4/1, -3/1)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (4/1, -3/1)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0015","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"caa1fbe11d595b5d8cd95ef0ba2d7dad94a7dbb603246b81f4634a79704aaa6b","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133015,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"2/3\",\"3/1\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=2/3, b=15/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(11/3, -7/3)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (11/3, -7/3)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0016","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8e26725596f314032c462a14eae914d071593ad5b5e7ad1363cdf5a2751a9bf","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133016,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"3/1\",\"4/1\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=3/1, b=-16/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(7/1, -1/1)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (7/1, -1/1)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0017","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"055f3479247264ec668b171adef5daad7b9d1b8586e0722fd919724055e60c6c","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133017,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"2/1\",\"17/5\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=4/2, b=17/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(27/5, -7/5)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (27/5, -7/5)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0018","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5189e40b7a700fea178408d9dcabadeee208ec5308a6a128ae3d987acb927f8e","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133018,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"5/3\",\"9/2\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=5/3, b=-18/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(37/6, -17/6)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (37/6, -17/6)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0019","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eec75ce5eb194aafad737c15e707674f8a00a20a68cfca0e6535d0737c0fa7d6","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133019,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"6/1\",\"19/5\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=6/1, b=19/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(49/5, 11/5)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (49/5, 11/5)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0020","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60ac4c06e88d6d0e47aab8e2bc0e330f4975db79c84cc21bddbf9881e351d2fa","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133020,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"7/2\",\"5/1\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=7/2, b=-20/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(17/2, -3/2)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (17/2, -3/2)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0021","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5f4f6f42d00f9dfb4cc96aa6160cc744780db4f089a94507f369b47708d4b66","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133021,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"8/3\",\"21/5\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=8/3, b=21/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(103/15, -23/15)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (103/15, -23/15)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0022","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c8cce3b1e00a689bf01d3791bd93b2fc953a79aab7f2c5a846a627b044ee3fd","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133022,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"9/1\",\"11/2\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=9/1, b=-22/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(29/2, 7/2)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (29/2, 7/2)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0023","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6aad401421bad19b4abcc58586f38d15ed3c4581de4f2d03c2b27b1f42e143b","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133023,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"5/1\",\"23/5\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=10/2, b=23/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(48/5, 2/5)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (48/5, 2/5)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0024","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bddd8a7422c7d4fbf39dba1a95f1cccbe86a00d4396972734d1eb08fe4a9f68d","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133024,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"11/3\",\"6/1\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=11/3, b=-24/-4. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(29/3, -7/3)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (29/3, -7/3)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0025","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3045bc914463205678f83588f52dfba98a1d907c2d4c7b37db4f592a09570e73","family_id":"qf-density13-rational-sum-difference-pair","seed":202610133025,"semantic_template":"qf-density13-rational-sum-difference-pair/[\"12/1\",\"5/1\"]"},"fields":{"title":"Exact Ordered Rational Sum and Difference","stem":"Exact rational representations have nonzero denominators; a=12/1, b=25/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(17/1, 7/1)","explanation":"Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (17/1, 7/1)."},"skills":["Exact Ordered Rational Sum and Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0026","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a149bae0b989c9cb9d2a9a3b8d55aaaa490b76377a68c7878961c9bd2ac4414","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133026,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"-3/1\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=-32/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(<, -3/1)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -3/1)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0027","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"568170f0a3fb732e601fa1433d52bff67b08cab2df70e0f9c8a05e6936e0bc5a","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133027,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"-11/4\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=29/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(<, -11/4)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -11/4)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0028","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e19019501b0f7b06f4a29af3095b95ba242bfc5ff0a29a612f390fd941cc3720","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133028,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"-5/2\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=-26/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(<, -5/2)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -5/2)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0029","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a9f6d7897fd09f188cdc7a0bc99bea774a128398b10ddc696cd73ed0080b3b0","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133029,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"-9/4\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=23/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(<, -9/4)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -9/4)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0030","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9d82923f6a8b61d1b96e4a95ef9d21b9879e3ccf2863454be053a0a6412c981","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133030,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"-2/1\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=-20/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(<, -2/1)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -2/1)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0031","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e3fef70502857bece90b312f0187d009938bf6ae7e78f5b445b7be40fa21e40","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133031,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"-7/4\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=17/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(<, -7/4)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -7/4)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0032","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29e3e8cfa44d6bc8878b78b2d187d613e51c7c952de35f2274f169e2293bf12b","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133032,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"-3/2\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=-14/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(<, -3/2)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -3/2)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0033","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b99be32662791e6c2c4a8b7d8ebfe2d43afef0d50d7de18ddadbe2855c1da90","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133033,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"-5/4\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=11/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(<, -5/4)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -5/4)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0034","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99aea08019c8a9b33c44246f9b3227531b1d699c13965ceb47c973100bb1c948","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133034,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"-1/1\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=-8/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(<, -1/1)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -1/1)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0035","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee92e8e382b94bb1c1197d2bfd0d59c407c1a1f6504fb05a9598da05aebbfdd0","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133035,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"-3/4\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=5/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(<, -3/4)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -3/4)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0036","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99d0fb14066c61b2b61cc8317398a0590156a202510699449f0dd0f8ff1fcd57","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133036,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"-1/2\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=-2/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(<, -1/2)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -1/2)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0037","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f70f23c98d01d908d824cf1f80a8a195f8bade36efb34e212ca23f1a7b6909c","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133037,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"-1/4\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=-1/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(<, -1/4)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (<, -1/4)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0038","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"171b6cfbb80d48e7beb4403508b1fea5f4153a5c7ced2960409850f3a59c5e9c","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133038,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"0/1\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=4/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(=, 0/1)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (=, 0/1)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0039","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac318f30e07653645747586eea60835badb3929034d47100291ccbaaa3cbc28d","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133039,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"1/4\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=-7/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(>, 1/4)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 1/4)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0040","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92172ea4c39bca546d8ff7dbfea08b1de74cf0d16b4b1eb5c596012aef2213f0","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133040,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"1/2\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=10/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(>, 1/2)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 1/2)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0041","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78ebac3410d71138a8f462b151e6fdc500a2ef85439827689d7cba1daeac5385","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133041,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"3/4\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=-13/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(>, 3/4)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 3/4)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0042","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc8297c567655e18f10b9e8122b60229780eb07d4ea4d45bc2023889894540d0","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133042,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"1/1\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=16/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(>, 1/1)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 1/1)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0043","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c19b6dc2a197ea827e223cb94da9f9958992550a879bfcde1880a666ac67be57","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133043,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"5/4\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=-19/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(>, 5/4)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 5/4)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0044","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f9dbc95fa3868d69902bf4ea1332248c544b5529df08f1cb2b5454bb3b5ee64","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133044,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"3/2\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=22/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(>, 3/2)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 3/2)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0045","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c1718001cf66f39997559b3a7f3a0bc6c5234839bb19ec459f98b552623de78","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133045,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"7/4\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=-25/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(>, 7/4)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 7/4)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0046","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b029016cd613d027b6803c746eda8f47a6793bcd886e3f822f146e1a7ba13e0","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133046,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"2/1\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=28/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(>, 2/1)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 2/1)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0047","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c903d63192d9461101a7f71c4f973de39a6483f4e034750c74f3747bc0591903","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133047,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"9/4\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=-31/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(>, 9/4)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 9/4)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0048","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be37493a8787e7814f3a27716f7fd35eee3c9136c4c5c4e4aa991c8164951204","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133048,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"5/2\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=34/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(>, 5/2)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 5/2)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0049","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e754d7412f15cdb4274f072d6b279b9e5058404ba400a2954c2f1646b419758d","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133049,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"11/4\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=-37/-12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(>, 11/4)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 11/4)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0050","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f31334adf9298462ded39a0af2c8656d26c8fcde5afae066621dd850a33c67a3","family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133050,"semantic_template":"qf-density13-rational-ordered-pair-compare/\"3/1\""},"fields":{"title":"Exact Rational Order and Signed Difference","stem":"Exact rational representations have nonzero denominators; a=40/12, b=1/3. Negative input denominators are allowed. Give (the <, =, or > relation of a to b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(>, 3/1)","explanation":"Determine order by the sign of the exact difference without approximate decimals; keep the signed difference. The exact result, retaining the requested order, is (>, 3/1)."},"skills":["Exact Rational Order and Signed Difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0051","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5aa64edef2fa3feee6a059543a1b6947a5b9f9c295dbaa0b4df76e5d9f6b030d","family_id":"qf-density13-rational-division-guard","seed":202610133051,"semantic_template":"qf-density13-rational-division-guard/[\"undefined\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=5/-2, b=0/-3. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(No, undefined, undefined)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (No, undefined, undefined)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0052","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e8a94617a4b05b704d54d2710c0981a0419bfa3f27f5d346a98c8d0486b65a6","family_id":"qf-density13-rational-division-guard","seed":202610133052,"semantic_template":"qf-density13-rational-division-guard/[\"zero_quotient\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=0/-7, b=2/-3. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 0/1, 0/1)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 0/1, 0/1)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0053","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8ef5d23b7ba6167ae820ed90a31ceeaa62f144e48b122eba2106f1d43b50c85","family_id":"qf-density13-rational-division-guard","seed":202610133053,"semantic_template":"qf-density13-rational-division-guard/[\"-1/1\",\"4/5\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=3/-3, b=4/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, -5/4, -1/1)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -5/4, -1/1)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0054","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f82d25812d81edf89b4d23bea05201adadaf40e15091ca12edff8c4ba497490","family_id":"qf-density13-rational-division-guard","seed":202610133054,"semantic_template":"qf-density13-rational-division-guard/[\"-4/3\",\"-5/4\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=4/-3, b=5/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 16/15, -4/3)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 16/15, -4/3)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0055","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8fb2f8393936fbb0d1141196c41970fed9e13a9f7c57bd7bb14afe47a3b5f76","family_id":"qf-density13-rational-division-guard","seed":202610133055,"semantic_template":"qf-density13-rational-division-guard/[\"-5/3\",\"6/5\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=5/-3, b=6/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, -25/18, -5/3)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -25/18, -5/3)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0056","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"587ced0c3f61d8a2670a0bd945b0765a1cf80174c25f4ba1d67636242233b115","family_id":"qf-density13-rational-division-guard","seed":202610133056,"semantic_template":"qf-density13-rational-division-guard/[\"-2/1\",\"-7/4\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=6/-3, b=7/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 8/7, -2/1)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 8/7, -2/1)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0057","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"145fc9d62721ec59f575d10f7ab5a2c8570f2d50a5cbedc14690cf9c7841c142","family_id":"qf-density13-rational-division-guard","seed":202610133057,"semantic_template":"qf-density13-rational-division-guard/[\"-7/3\",\"8/5\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=7/-3, b=8/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, -35/24, -7/3)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -35/24, -7/3)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0058","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdd7c3754ee5df9ad6b554a636b31bf170d1689c94d016a0e9941d6da9eb9d3a","family_id":"qf-density13-rational-division-guard","seed":202610133058,"semantic_template":"qf-density13-rational-division-guard/[\"-8/3\",\"-9/4\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=8/-3, b=9/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 32/27, -8/3)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 32/27, -8/3)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0059","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76831bfa1afb151bc70680830801d3e124fde7086c5500fd97c8f60255a8ff77","family_id":"qf-density13-rational-division-guard","seed":202610133059,"semantic_template":"qf-density13-rational-division-guard/[\"-3/1\",\"2/1\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=9/-3, b=10/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, -3/2, -3/1)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -3/2, -3/1)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0060","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96fd03971b09a4ebc11ee5256b39ffeeb5e86e27814ceb4253820905e707967a","family_id":"qf-density13-rational-division-guard","seed":202610133060,"semantic_template":"qf-density13-rational-division-guard/[\"-10/3\",\"-11/4\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=10/-3, b=11/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 40/33, -10/3)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 40/33, -10/3)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0061","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87a1ca264f1b5c37b7d4fb72b890e3cf8324914e8784d2473dffa5ba60d970c2","family_id":"qf-density13-rational-division-guard","seed":202610133061,"semantic_template":"qf-density13-rational-division-guard/[\"-11/3\",\"12/5\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=11/-3, b=12/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, -55/36, -11/3)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -55/36, -11/3)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0062","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d441327895554595d53df8a33ab52a2a243f4cd30f49935c339cd15471acb0f","family_id":"qf-density13-rational-division-guard","seed":202610133062,"semantic_template":"qf-density13-rational-division-guard/[\"-4/1\",\"-13/4\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=12/-3, b=13/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 16/13, -4/1)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 16/13, -4/1)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0063","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b65487bcb3ae5de2a5cef82b5733d7470cbffd317e71b45c48595469789bd3a8","family_id":"qf-density13-rational-division-guard","seed":202610133063,"semantic_template":"qf-density13-rational-division-guard/[\"-13/3\",\"14/5\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=13/-3, b=14/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, -65/42, -13/3)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -65/42, -13/3)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0064","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf834b5d8a10ce0868fb6c50cfc1b27567b0f16c50f96345dd907cc9bbcadd88","family_id":"qf-density13-rational-division-guard","seed":202610133064,"semantic_template":"qf-density13-rational-division-guard/[\"-14/3\",\"-15/4\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=14/-3, b=15/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 56/45, -14/3)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 56/45, -14/3)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0065","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d9732416ae2fad776d78159a61409e6c18af5b2504403400b8b1b9dd4cc86eb","family_id":"qf-density13-rational-division-guard","seed":202610133065,"semantic_template":"qf-density13-rational-division-guard/[\"-5/1\",\"16/5\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=15/-3, b=16/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, -25/16, -5/1)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -25/16, -5/1)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0066","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bd97b4550a176b275f2d5b19cd0db205571e18e8c43748ac4afafd707cdcca4","family_id":"qf-density13-rational-division-guard","seed":202610133066,"semantic_template":"qf-density13-rational-division-guard/[\"-16/3\",\"-17/4\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=16/-3, b=17/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 64/51, -16/3)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 64/51, -16/3)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0067","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d344b642240ada2eadcb120e18ce675512e81a12f7a66b94942426962fa5b718","family_id":"qf-density13-rational-division-guard","seed":202610133067,"semantic_template":"qf-density13-rational-division-guard/[\"-17/3\",\"18/5\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=17/-3, b=18/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, -85/54, -17/3)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -85/54, -17/3)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0068","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9002b54900d23a2b523a74e5f544b7e2565f7692943929ea2a534798346b96b6","family_id":"qf-density13-rational-division-guard","seed":202610133068,"semantic_template":"qf-density13-rational-division-guard/[\"-6/1\",\"-19/4\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=18/-3, b=19/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 24/19, -6/1)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 24/19, -6/1)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0069","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2305a408f6f5b8cd85f331b8fc51975895d7cd8bf992241aea9af802795827b1","family_id":"qf-density13-rational-division-guard","seed":202610133069,"semantic_template":"qf-density13-rational-division-guard/[\"-19/3\",\"4/1\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=19/-3, b=20/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, -19/12, -19/3)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -19/12, -19/3)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0070","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb0a517f101fe03ff4ecb9f99af5a9a15dee389a5b159d5e2232f0edf6bf05c1","family_id":"qf-density13-rational-division-guard","seed":202610133070,"semantic_template":"qf-density13-rational-division-guard/[\"-20/3\",\"-21/4\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=20/-3, b=21/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 80/63, -20/3)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 80/63, -20/3)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0071","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f682899d239210321dee48645f2c6854e4cefd8bbe08d4dbdd9b7b293a51f6a8","family_id":"qf-density13-rational-division-guard","seed":202610133071,"semantic_template":"qf-density13-rational-division-guard/[\"-7/1\",\"22/5\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=21/-3, b=22/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, -35/22, -7/1)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -35/22, -7/1)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0072","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbf6b205289877b9fb14076a1f5f95f476ca5f942c75f3bac5b29acc31d9f3d0","family_id":"qf-density13-rational-division-guard","seed":202610133072,"semantic_template":"qf-density13-rational-division-guard/[\"-22/3\",\"-23/4\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=22/-3, b=23/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 88/69, -22/3)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 88/69, -22/3)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0073","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77f9a86e9126dffd3bc163074e35b9c3be6b32907b5f16a1a7441c32ff10dca3","family_id":"qf-density13-rational-division-guard","seed":202610133073,"semantic_template":"qf-density13-rational-division-guard/[\"-23/3\",\"24/5\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=23/-3, b=24/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, -115/72, -23/3)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -115/72, -23/3)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0074","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91e048d75d37f4ee7be3d782f03d0451c3176d977c393314940ac35c42e3a5b5","family_id":"qf-density13-rational-division-guard","seed":202610133074,"semantic_template":"qf-density13-rational-division-guard/[\"-8/1\",\"-25/4\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=24/-3, b=25/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, 32/25, -8/1)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 32/25, -8/1)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0075","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3faeb071a6d42bea8296267dfb17bb88274266db66c0de99869bd6233ac14d4","family_id":"qf-density13-rational-division-guard","seed":202610133075,"semantic_template":"qf-density13-rational-division-guard/[\"-25/3\",\"26/5\"]"},"fields":{"title":"Rational Division Eligibility and Multiplication Back","stem":"Exact rational representations have nonzero denominators; a=25/-3, b=26/5. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(Yes, -125/78, -25/3)","explanation":"Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, -125/78, -25/3)."},"skills":["Rational Division Eligibility and Multiplication Back"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0076","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc5327a54861709f8e03c4678d612201f57d19f700ce6a7c7c44042b6544cb27","family_id":"qf-density13-rational-target-missing-addend","seed":202610133076,"semantic_template":"qf-density13-rational-target-missing-addend/\"-4/1\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -23/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-4/1, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-4/1, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0077","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50f520e8dab8b0f22dfe2b387d27d8b42025e0b8791ad4d31ce8eddf6f2afba7","family_id":"qf-density13-rational-target-missing-addend","seed":202610133077,"semantic_template":"qf-density13-rational-target-missing-addend/\"-11/3\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 21/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-11/3, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-11/3, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0078","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99bfe292bc4d73482281462fc87c56c2c7bf1cda83f9389e4103f730a044dc6c","family_id":"qf-density13-rational-target-missing-addend","seed":202610133078,"semantic_template":"qf-density13-rational-target-missing-addend/\"-10/3\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -19/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-10/3, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-10/3, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0079","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c4cc1cf22c4621c9f993ef820e42b93eba250e4053c0f9f4c4f183ff37b4cb9","family_id":"qf-density13-rational-target-missing-addend","seed":202610133079,"semantic_template":"qf-density13-rational-target-missing-addend/\"-3/1\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 17/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-3/1, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-3/1, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0080","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88a75ee11cf2be7f94503d26017a41ed05af2aafa1b61c2541603d050e107d14","family_id":"qf-density13-rational-target-missing-addend","seed":202610133080,"semantic_template":"qf-density13-rational-target-missing-addend/\"-8/3\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -15/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-8/3, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-8/3, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0081","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf64c0a0c6611bee78c2f5f2bddb172aaf45fd763ba5ab8467b29950739b67cc","family_id":"qf-density13-rational-target-missing-addend","seed":202610133081,"semantic_template":"qf-density13-rational-target-missing-addend/\"-7/3\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 13/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-7/3, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-7/3, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0082","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3bd4866f60e33275e068529098187b2798f529c10014f753e9080d16b85a4a6","family_id":"qf-density13-rational-target-missing-addend","seed":202610133082,"semantic_template":"qf-density13-rational-target-missing-addend/\"-2/1\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -11/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-2/1, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-2/1, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0083","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31fc0a252c51eb54f09aefc15ad55874a9ee42238ad030093133c35866b5b574","family_id":"qf-density13-rational-target-missing-addend","seed":202610133083,"semantic_template":"qf-density13-rational-target-missing-addend/\"-5/3\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 9/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-5/3, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-5/3, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0084","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97a346231e833498507c47851b265fa0743c6fd540f84250a80b007d35c8824d","family_id":"qf-density13-rational-target-missing-addend","seed":202610133084,"semantic_template":"qf-density13-rational-target-missing-addend/\"-4/3\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -7/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-4/3, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-4/3, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0085","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f7f328ce8cebfa807e8f407d585b331239694995494856f59641d999b13764a","family_id":"qf-density13-rational-target-missing-addend","seed":202610133085,"semantic_template":"qf-density13-rational-target-missing-addend/\"-1/1\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 5/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/1, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-1/1, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0086","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"548e88255bdaec9f90f695cd40639536e527d42a9d8d8e66487ab19cbdd18444","family_id":"qf-density13-rational-target-missing-addend","seed":202610133086,"semantic_template":"qf-density13-rational-target-missing-addend/\"-2/3\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -3/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-2/3, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-2/3, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0087","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3381af09cfed4d48cd17b05960dab2dbf48a34fee02581dea7624fa32bdb253","family_id":"qf-density13-rational-target-missing-addend","seed":202610133087,"semantic_template":"qf-density13-rational-target-missing-addend/\"-1/3\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 1/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1/3, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-1/3, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0088","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b85a4dd9f2757c7deb2f01f34a2c8041b3cf481a6f7abfffcd431be6412562c","family_id":"qf-density13-rational-target-missing-addend","seed":202610133088,"semantic_template":"qf-density13-rational-target-missing-addend/\"0/1\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 1/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (0/1, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0089","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da331dc23490800b70c52f8a5f5791c1bc05ab10a7311ebd84ab184c78b39557","family_id":"qf-density13-rational-target-missing-addend","seed":202610133089,"semantic_template":"qf-density13-rational-target-missing-addend/\"1/3\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -3/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/3, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (1/3, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0090","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d85bebc7431cf908d336c4197ae21f8250937f689bdd125df8dbbfd31cdf6702","family_id":"qf-density13-rational-target-missing-addend","seed":202610133090,"semantic_template":"qf-density13-rational-target-missing-addend/\"2/3\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 5/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(2/3, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (2/3, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0091","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f8d2e9cdfb712c04168bab28ed39c7c15f13535b978fd65e8b011c7e9e614a8","family_id":"qf-density13-rational-target-missing-addend","seed":202610133091,"semantic_template":"qf-density13-rational-target-missing-addend/\"1/1\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -7/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/1, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (1/1, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0092","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c35ddf40f96824e39bb102e3823ed4e5afdd8bd9e641ff17f50cbfa90c1f90e","family_id":"qf-density13-rational-target-missing-addend","seed":202610133092,"semantic_template":"qf-density13-rational-target-missing-addend/\"4/3\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 9/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(4/3, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (4/3, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0093","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b39f6a4d1fcaeeeeb745c8c2d1f46da3577cf40fd35e7c9e7e9a085542de0daa","family_id":"qf-density13-rational-target-missing-addend","seed":202610133093,"semantic_template":"qf-density13-rational-target-missing-addend/\"5/3\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -11/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(5/3, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (5/3, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0094","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"744535daa19ae465419390461edbc0953a77165fcdee45a0b7e49d0b8fde1122","family_id":"qf-density13-rational-target-missing-addend","seed":202610133094,"semantic_template":"qf-density13-rational-target-missing-addend/\"2/1\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 13/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(2/1, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (2/1, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0095","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d208c6c02e8f4752d25587d03ceae097485340af3c0417c1664ccb7bc2a1b4ec","family_id":"qf-density13-rational-target-missing-addend","seed":202610133095,"semantic_template":"qf-density13-rational-target-missing-addend/\"7/3\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -15/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(7/3, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (7/3, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0096","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4504129679cc1f4bee21df9022564626344ddb7bcde7e6549d71c982de372376","family_id":"qf-density13-rational-target-missing-addend","seed":202610133096,"semantic_template":"qf-density13-rational-target-missing-addend/\"8/3\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 17/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(8/3, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (8/3, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0097","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52a47bfe56808cf1171e0446276c3f2da3e690687e114bc5ea0e7447c1de4715","family_id":"qf-density13-rational-target-missing-addend","seed":202610133097,"semantic_template":"qf-density13-rational-target-missing-addend/\"3/1\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -19/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(3/1, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (3/1, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0098","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba04ad60cff0d32b209de2759d6c93a0f0f2cf5f9676fa2eb550da81faeddad2","family_id":"qf-density13-rational-target-missing-addend","seed":202610133098,"semantic_template":"qf-density13-rational-target-missing-addend/\"10/3\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 21/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(10/3, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (10/3, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0099","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aac283358e725f777b07b44cb57d8bacb159acdd362367105cbddabfd135218d","family_id":"qf-density13-rational-target-missing-addend","seed":202610133099,"semantic_template":"qf-density13-rational-target-missing-addend/\"11/3\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -23/-6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(11/3, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (11/3, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r03-rational-number-0100","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7026833ccc6e19ec15781fcbfa72ccab0110a7a4e8f2d5ca7024a42cb15733bd","family_id":"qf-density13-rational-target-missing-addend","seed":202610133100,"semantic_template":"qf-density13-rational-target-missing-addend/\"4/1\""},"fields":{"title":"Missing Rational Addend and Substitution Defect","stem":"Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is 25/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(4/1, 0/1)","explanation":"Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (4/1, 0/1)."},"skills":["Missing Rational Addend and Substitution Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}