{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r02-ratio-proportion-0001","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81810402d3bdccfe0e55e2f24afefcf2f230dd57c8a1ca3a959e703ce996b4ac","family_id":"qf-density15-ratio-forward-reverse","seed":202610152001,"semantic_template":"qf-density15-ratio-forward-reverse/\"0/1\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=0, b=-5≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(0/1, No, undefined, undefined)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (0/1, No, undefined, undefined)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0002","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df1cd7aa32abde91d26fc783c2fc5e8d43bdcfa2e83b0d28777d61916c7fea54","family_id":"qf-density15-ratio-forward-reverse","seed":202610152002,"semantic_template":"qf-density15-ratio-forward-reverse/\"-1/2\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=1, b=-2≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(-1/2, Yes, -2/1, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (-1/2, Yes, -2/1, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0003","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"459938e594b627611bd7b428a97424487c355f980bca593a0ee04882bc88f039","family_id":"qf-density15-ratio-forward-reverse","seed":202610152003,"semantic_template":"qf-density15-ratio-forward-reverse/\"1/3\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=1, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(1/3, Yes, 3/1, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (1/3, Yes, 3/1, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0004","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"395450fa556c3265a85efef93cbe6cd51f118cd9b33048c7c90b60457ba90c70","family_id":"qf-density15-ratio-forward-reverse","seed":202610152004,"semantic_template":"qf-density15-ratio-forward-reverse/\"2/3\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=2, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(2/3, Yes, 3/2, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (2/3, Yes, 3/2, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0005","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59e6e8bc8a140438f8948d45f554a8087453c4bffd5e721fe8af9a9ee2b98f87","family_id":"qf-density15-ratio-forward-reverse","seed":202610152005,"semantic_template":"qf-density15-ratio-forward-reverse/\"1/1\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=3, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(1/1, Yes, 1/1, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (1/1, Yes, 1/1, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0006","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"541ec552a501353c1d730bc2b49d6aecbb3c91cb22f7457624975d58ed6e5cab","family_id":"qf-density15-ratio-forward-reverse","seed":202610152006,"semantic_template":"qf-density15-ratio-forward-reverse/\"4/3\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=4, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(4/3, Yes, 3/4, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (4/3, Yes, 3/4, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0007","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39c7035c71a82a201c83a2569b3dbbd663bbc4e3c535f361b9bc3a8ab1111fba","family_id":"qf-density15-ratio-forward-reverse","seed":202610152007,"semantic_template":"qf-density15-ratio-forward-reverse/\"5/3\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=5, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(5/3, Yes, 3/5, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (5/3, Yes, 3/5, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0008","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8371d9f45be5f9da6299954b70c2113d5d28a0cd46cb6ae16179af8833c8d68e","family_id":"qf-density15-ratio-forward-reverse","seed":202610152008,"semantic_template":"qf-density15-ratio-forward-reverse/\"2/1\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=6, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(2/1, Yes, 1/2, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (2/1, Yes, 1/2, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0009","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3657030c07bfff411962414253f71923836a047f27d15025778fec9ee8d04fdc","family_id":"qf-density15-ratio-forward-reverse","seed":202610152009,"semantic_template":"qf-density15-ratio-forward-reverse/\"7/3\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=7, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(7/3, Yes, 3/7, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (7/3, Yes, 3/7, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0010","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11cee6753bc77042637663cc2ecee67252c515238f035d1ab29c1a4f73d4738d","family_id":"qf-density15-ratio-forward-reverse","seed":202610152010,"semantic_template":"qf-density15-ratio-forward-reverse/\"8/3\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=8, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(8/3, Yes, 3/8, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (8/3, Yes, 3/8, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0011","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edcb6d47aad60d91949ff2647f5bf8b124d2c1cc12bf1e7cd4a64185eb766040","family_id":"qf-density15-ratio-forward-reverse","seed":202610152011,"semantic_template":"qf-density15-ratio-forward-reverse/\"3/1\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=9, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(3/1, Yes, 1/3, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (3/1, Yes, 1/3, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0012","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d6c02ebefd9d44fd32e46e107791c0971c77658a9995612b9908d0aad0a1c23","family_id":"qf-density15-ratio-forward-reverse","seed":202610152012,"semantic_template":"qf-density15-ratio-forward-reverse/\"10/3\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=10, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(10/3, Yes, 3/10, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (10/3, Yes, 3/10, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0013","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afd1b0d054af1a3d270f602549741c098f8bd00bd04a97ac6da217cedb128b89","family_id":"qf-density15-ratio-forward-reverse","seed":202610152013,"semantic_template":"qf-density15-ratio-forward-reverse/\"11/3\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=11, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(11/3, Yes, 3/11, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (11/3, Yes, 3/11, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0014","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05969fd8ba2e9ec44830b7dd0a4ae856f803a48acae627c9da2073018b466adf","family_id":"qf-density15-ratio-forward-reverse","seed":202610152014,"semantic_template":"qf-density15-ratio-forward-reverse/\"4/1\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=12, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(4/1, Yes, 1/4, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (4/1, Yes, 1/4, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0015","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a31449f2b20d22f18b3ca370a37e4fe1a1a3ea4b726648b7e059f228d00bf1c","family_id":"qf-density15-ratio-forward-reverse","seed":202610152015,"semantic_template":"qf-density15-ratio-forward-reverse/\"13/3\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=13, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(13/3, Yes, 3/13, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (13/3, Yes, 3/13, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0016","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36863262824cc393f5614b9f5c9794af3c799235810af50de94e0f6ff518f69c","family_id":"qf-density15-ratio-forward-reverse","seed":202610152016,"semantic_template":"qf-density15-ratio-forward-reverse/\"14/3\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=14, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(14/3, Yes, 3/14, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (14/3, Yes, 3/14, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0017","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"160f4608bfa41f330eccafb3866c65c3d13b284418905eb31ec69ff480772ec7","family_id":"qf-density15-ratio-forward-reverse","seed":202610152017,"semantic_template":"qf-density15-ratio-forward-reverse/\"5/1\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=15, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(5/1, Yes, 1/5, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (5/1, Yes, 1/5, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0018","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5d896f7880e1af02f5a74bbdae03d5635d22ef93713a1e8ae071ef877c552ca","family_id":"qf-density15-ratio-forward-reverse","seed":202610152018,"semantic_template":"qf-density15-ratio-forward-reverse/\"16/3\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=16, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(16/3, Yes, 3/16, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (16/3, Yes, 3/16, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0019","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2065680d541e53ca4acc89a0f635c9e1cf250aca03e0cda7cbd334328142c9e5","family_id":"qf-density15-ratio-forward-reverse","seed":202610152019,"semantic_template":"qf-density15-ratio-forward-reverse/\"17/3\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=17, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(17/3, Yes, 3/17, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (17/3, Yes, 3/17, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0020","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6124ca542590000dbc1923cefe8fd67a9cf8f0bda9e770df29d66f078538870","family_id":"qf-density15-ratio-forward-reverse","seed":202610152020,"semantic_template":"qf-density15-ratio-forward-reverse/\"6/1\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=18, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(6/1, Yes, 1/6, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (6/1, Yes, 1/6, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0021","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f14d4074a793ac71a57792b91cee843f849cd362225c5fad0893f98a6e3e8e80","family_id":"qf-density15-ratio-forward-reverse","seed":202610152021,"semantic_template":"qf-density15-ratio-forward-reverse/\"19/3\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=19, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(19/3, Yes, 3/19, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (19/3, Yes, 3/19, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0022","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64ec46b61241f7066de4cc5cc29e3f1a5e80992ad3e6c82ce4616c6c077667b2","family_id":"qf-density15-ratio-forward-reverse","seed":202610152022,"semantic_template":"qf-density15-ratio-forward-reverse/\"20/3\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=20, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(20/3, Yes, 3/20, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (20/3, Yes, 3/20, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0023","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bdac1ae22fa1fd346ce7dd4edf20a57092dc011c2035de4a95c59068688a17b","family_id":"qf-density15-ratio-forward-reverse","seed":202610152023,"semantic_template":"qf-density15-ratio-forward-reverse/\"7/1\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=21, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(7/1, Yes, 1/7, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (7/1, Yes, 1/7, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0024","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1b2e7e15c56a09787518311174107e538770bea7114ade30ec7537457a16b6d","family_id":"qf-density15-ratio-forward-reverse","seed":202610152024,"semantic_template":"qf-density15-ratio-forward-reverse/\"22/3\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=22, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(22/3, Yes, 3/22, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (22/3, Yes, 3/22, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0025","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08040cd20d7d62eb6dce62536b96677ebccae4b3485b1b75d4f73c77b5f9e628","family_id":"qf-density15-ratio-forward-reverse","seed":202610152025,"semantic_template":"qf-density15-ratio-forward-reverse/\"23/3\""},"fields":{"title":"Forward Ratio and Independent Reverse Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=23, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).","answer":{"kind":"expression","value":"(23/3, Yes, 3/23, 1/1)","explanation":"The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (23/3, Yes, 3/23, 1/1)."},"skills":["Forward Ratio and Independent Reverse Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0026","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5456b4f93dbd472e1ba638b76dc0d81b04eb2f6eecf86124999cdf792444f867","family_id":"qf-density15-ratio-complete-point-law","seed":202610152026,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"0/1\",\"1/1\"],[\"0/1\",\"1/1\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=1/1, complete ordered pairs (x,y)=[[0, 0], [1, 1], [2, 2]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 0/1, 0/1], [Yes, Yes, Yes], [undefined, 1/1, 1/1])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 0/1, 0/1], [Yes, Yes, Yes], [undefined, 1/1, 1/1])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0027","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1baab4fc176899a335546cc8d2596eb98bb2e3184fbf396b65ec3ff3f54dae21","family_id":"qf-density15-ratio-complete-point-law","seed":202610152027,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"1/1\",\"undefined\"],[\"0/1\",\"1/1\"],[\"0/1\",\"1/1\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=1/1, complete ordered pairs (x,y)=[[0, 1], [1, 1], [2, 2]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([1/1, 0/1, 0/1], [No, Yes, Yes], [undefined, 1/1, 1/1])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([1/1, 0/1, 0/1], [No, Yes, Yes], [undefined, 1/1, 1/1])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0028","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ea1ed746628c6f4f6b4a0581e97fbad883bed4a8691e3954fcd76de23b18d36","family_id":"qf-density15-ratio-complete-point-law","seed":202610152028,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"7/1\",\"2/1\"],[\"9/1\",\"-1/2\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-10/2, complete ordered pairs (x,y)=[[0, 0], [1, 2], [2, -1]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 7/1, 9/1], [Yes, No, No], [undefined, 2/1, -1/2])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 7/1, 9/1], [Yes, No, No], [undefined, 2/1, -1/2])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0029","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd5ca72c5e309a7a8c47b7bf7f666203f01cded11d6e1d1c4ee8c5271220e22d","family_id":"qf-density15-ratio-complete-point-law","seed":202610152029,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"15/2\",\"3/1\"],[\"7/1\",\"-1/1\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-9/2, complete ordered pairs (x,y)=[[0, 0], [1, 3], [2, -2]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 15/2, 7/1], [Yes, No, No], [undefined, 3/1, -1/1])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 15/2, 7/1], [Yes, No, No], [undefined, 3/1, -1/1])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0030","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8079029244d3e0eb0e7d2499c11834d2cae0692a0f3afd08343a8ca5d4472b6","family_id":"qf-density15-ratio-complete-point-law","seed":202610152030,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"8/1\",\"4/1\"],[\"5/1\",\"-3/2\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-8/2, complete ordered pairs (x,y)=[[0, 0], [1, 4], [2, -3]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 8/1, 5/1], [Yes, No, No], [undefined, 4/1, -3/2])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 8/1, 5/1], [Yes, No, No], [undefined, 4/1, -3/2])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0031","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95b289fd524f3c2b97d2882b048801dd3ac0541fc65973f2052ce99e39a73b1f","family_id":"qf-density15-ratio-complete-point-law","seed":202610152031,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"17/2\",\"5/1\"],[\"3/1\",\"-2/1\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-7/2, complete ordered pairs (x,y)=[[0, 0], [1, 5], [2, -4]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 17/2, 3/1], [Yes, No, No], [undefined, 5/1, -2/1])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 17/2, 3/1], [Yes, No, No], [undefined, 5/1, -2/1])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0032","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f392bc78cfc968200462ed8a2723f2b700db6da7ecfd369a58a22addeee8f886","family_id":"qf-density15-ratio-complete-point-law","seed":202610152032,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"9/1\",\"6/1\"],[\"1/1\",\"-5/2\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-6/2, complete ordered pairs (x,y)=[[0, 0], [1, 6], [2, -5]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 9/1, 1/1], [Yes, No, No], [undefined, 6/1, -5/2])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 9/1, 1/1], [Yes, No, No], [undefined, 6/1, -5/2])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0033","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff0d5e0c9e181981034c3a12e08026b908e1527a6ff0f442a1b45edd35c08ee3","family_id":"qf-density15-ratio-complete-point-law","seed":202610152033,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"19/2\",\"7/1\"],[\"-1/1\",\"-3/1\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-5/2, complete ordered pairs (x,y)=[[0, 0], [1, 7], [2, -6]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 19/2, -1/1], [Yes, No, No], [undefined, 7/1, -3/1])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 19/2, -1/1], [Yes, No, No], [undefined, 7/1, -3/1])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0034","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f107d222ec5db4c8a2a1fc5027d01468a303a62d65de6d70447d299ed32b7d1","family_id":"qf-density15-ratio-complete-point-law","seed":202610152034,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"10/1\",\"8/1\"],[\"-3/1\",\"-7/2\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-4/2, complete ordered pairs (x,y)=[[0, 0], [1, 8], [2, -7]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 10/1, -3/1], [Yes, No, No], [undefined, 8/1, -7/2])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 10/1, -3/1], [Yes, No, No], [undefined, 8/1, -7/2])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0035","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab5d4612f36c2b13509591ba39e1ab52a0df32a34aec779269de8cf21196fe41","family_id":"qf-density15-ratio-complete-point-law","seed":202610152035,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"21/2\",\"9/1\"],[\"-5/1\",\"-4/1\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-3/2, complete ordered pairs (x,y)=[[0, 0], [1, 9], [2, -8]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 21/2, -5/1], [Yes, No, No], [undefined, 9/1, -4/1])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 21/2, -5/1], [Yes, No, No], [undefined, 9/1, -4/1])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0036","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e43b97590e83833769bfa8da95d6f1e38b1edeb58c7a05afb60562ad1e53165","family_id":"qf-density15-ratio-complete-point-law","seed":202610152036,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"11/1\",\"10/1\"],[\"-7/1\",\"-9/2\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-2/2, complete ordered pairs (x,y)=[[0, 0], [1, 10], [2, -9]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 11/1, -7/1], [Yes, No, No], [undefined, 10/1, -9/2])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 11/1, -7/1], [Yes, No, No], [undefined, 10/1, -9/2])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0037","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdd21f3db1b3f585afbc6fa293a00df06a8bab9d97435279bab7a1f894a4d701","family_id":"qf-density15-ratio-complete-point-law","seed":202610152037,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"23/2\",\"11/1\"],[\"-9/1\",\"-5/1\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=-1/2, complete ordered pairs (x,y)=[[0, 0], [1, 11], [2, -10]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 23/2, -9/1], [Yes, No, No], [undefined, 11/1, -5/1])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 23/2, -9/1], [Yes, No, No], [undefined, 11/1, -5/1])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0038","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"231a821d66cb36e5080094692ce0b6f3ff3627dd6b9212e3d8daa19a646a68a7","family_id":"qf-density15-ratio-complete-point-law","seed":202610152038,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"12/1\",\"12/1\"],[\"-11/1\",\"-11/2\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=0/2, complete ordered pairs (x,y)=[[0, 0], [1, 12], [2, -11]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 12/1, -11/1], [Yes, No, No], [undefined, 12/1, -11/2])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 12/1, -11/1], [Yes, No, No], [undefined, 12/1, -11/2])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0039","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"701a5f3789f561eb2a674db053b8015450402b5af41bf78a7d7d967166be50b1","family_id":"qf-density15-ratio-complete-point-law","seed":202610152039,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"25/2\",\"13/1\"],[\"-13/1\",\"-6/1\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=1/2, complete ordered pairs (x,y)=[[0, 0], [1, 13], [2, -12]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 25/2, -13/1], [Yes, No, No], [undefined, 13/1, -6/1])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 25/2, -13/1], [Yes, No, No], [undefined, 13/1, -6/1])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0040","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcacfb36b9b870f50951ab57a88618ed3a5ea733b8295ce84756813c251e5b29","family_id":"qf-density15-ratio-complete-point-law","seed":202610152040,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"13/1\",\"14/1\"],[\"-15/1\",\"-13/2\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=2/2, complete ordered pairs (x,y)=[[0, 0], [1, 14], [2, -13]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 13/1, -15/1], [Yes, No, No], [undefined, 14/1, -13/2])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 13/1, -15/1], [Yes, No, No], [undefined, 14/1, -13/2])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0041","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff7eb26f3c8ad4366488bc1b0ccafcf92d5ae7d298e8e7faa012f93a310ac065","family_id":"qf-density15-ratio-complete-point-law","seed":202610152041,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"27/2\",\"15/1\"],[\"-17/1\",\"-7/1\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=3/2, complete ordered pairs (x,y)=[[0, 0], [1, 15], [2, -14]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 27/2, -17/1], [Yes, No, No], [undefined, 15/1, -7/1])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 27/2, -17/1], [Yes, No, No], [undefined, 15/1, -7/1])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0042","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"969aa16a36bd64c0a2d6df5f564d66e0b1bcf88f374342bf4d9e59a3c3e22c9f","family_id":"qf-density15-ratio-complete-point-law","seed":202610152042,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"14/1\",\"16/1\"],[\"-19/1\",\"-15/2\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=4/2, complete ordered pairs (x,y)=[[0, 0], [1, 16], [2, -15]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 14/1, -19/1], [Yes, No, No], [undefined, 16/1, -15/2])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 14/1, -19/1], [Yes, No, No], [undefined, 16/1, -15/2])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0043","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8faeca05622e3261e7ca20f4e274843e2043af98b9d2ab7c841f26a345d2fe8","family_id":"qf-density15-ratio-complete-point-law","seed":202610152043,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"29/2\",\"17/1\"],[\"-21/1\",\"-8/1\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=5/2, complete ordered pairs (x,y)=[[0, 0], [1, 17], [2, -16]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 29/2, -21/1], [Yes, No, No], [undefined, 17/1, -8/1])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 29/2, -21/1], [Yes, No, No], [undefined, 17/1, -8/1])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0044","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"caa3ffb3f633fbb97cce214d1f0c3698573d5dc98b1b37ed39d59f12fbdbd98c","family_id":"qf-density15-ratio-complete-point-law","seed":202610152044,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"15/1\",\"18/1\"],[\"-23/1\",\"-17/2\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=6/2, complete ordered pairs (x,y)=[[0, 0], [1, 18], [2, -17]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 15/1, -23/1], [Yes, No, No], [undefined, 18/1, -17/2])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 15/1, -23/1], [Yes, No, No], [undefined, 18/1, -17/2])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0045","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6cc74e19f9594b05b35d95bbf19dc259b174aa08fa9cf5ed73091a5925f51ea","family_id":"qf-density15-ratio-complete-point-law","seed":202610152045,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"31/2\",\"19/1\"],[\"-25/1\",\"-9/1\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=7/2, complete ordered pairs (x,y)=[[0, 0], [1, 19], [2, -18]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 31/2, -25/1], [Yes, No, No], [undefined, 19/1, -9/1])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 31/2, -25/1], [Yes, No, No], [undefined, 19/1, -9/1])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0046","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da6e8c8079c86534bddd506d98da81ec73490cab7f9513a8c5ec4d6c2f2bc42c","family_id":"qf-density15-ratio-complete-point-law","seed":202610152046,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"16/1\",\"20/1\"],[\"-27/1\",\"-19/2\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=8/2, complete ordered pairs (x,y)=[[0, 0], [1, 20], [2, -19]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 16/1, -27/1], [Yes, No, No], [undefined, 20/1, -19/2])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 16/1, -27/1], [Yes, No, No], [undefined, 20/1, -19/2])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0047","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb0f049d8b7d1d6aa9d66f91eaac10bc1655598cf9b9b1853d04952f73993e98","family_id":"qf-density15-ratio-complete-point-law","seed":202610152047,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"33/2\",\"21/1\"],[\"-29/1\",\"-10/1\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=9/2, complete ordered pairs (x,y)=[[0, 0], [1, 21], [2, -20]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 33/2, -29/1], [Yes, No, No], [undefined, 21/1, -10/1])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 33/2, -29/1], [Yes, No, No], [undefined, 21/1, -10/1])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0048","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26cc5e3e6c8547e97ec733245a74d6cb3740a009db2add41af8143042c35aaa6","family_id":"qf-density15-ratio-complete-point-law","seed":202610152048,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"17/1\",\"22/1\"],[\"-31/1\",\"-21/2\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=10/2, complete ordered pairs (x,y)=[[0, 0], [1, 22], [2, -21]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 17/1, -31/1], [Yes, No, No], [undefined, 22/1, -21/2])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 17/1, -31/1], [Yes, No, No], [undefined, 22/1, -21/2])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0049","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e007cee464404212206e0cb5bb117596a6fa597b4c7c0352497908825265e389","family_id":"qf-density15-ratio-complete-point-law","seed":202610152049,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"35/2\",\"23/1\"],[\"-33/1\",\"-11/1\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=11/2, complete ordered pairs (x,y)=[[0, 0], [1, 23], [2, -22]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 35/2, -33/1], [Yes, No, No], [undefined, 23/1, -11/1])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 35/2, -33/1], [Yes, No, No], [undefined, 23/1, -11/1])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0050","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6925e8b6d8b51ade4eebfc7a06559aa9b776320821f4117f6cb8d76640d438dd","family_id":"qf-density15-ratio-complete-point-law","seed":202610152050,"semantic_template":"qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"18/1\",\"24/1\"],[\"-35/1\",\"-23/2\"]]"},"fields":{"title":"Complete Three-Point Proportion Law and Ratio Values","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=12/2, complete ordered pairs (x,y)=[[0, 0], [1, 24], [2, -23]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.","answer":{"kind":"expression","value":"([0/1, 18/1, -35/1], [Yes, No, No], [undefined, 24/1, -23/2])","explanation":"Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 18/1, -35/1], [Yes, No, No], [undefined, 24/1, -23/2])."},"skills":["Complete Three-Point Proportion Law and Ratio Values"],"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-20261011-advance-20261015-r02-ratio-proportion-0051","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6f8e49948c11abd2c8f9a8d5344142e24c3dac221655733a8c46cd7e204d5b2","family_id":"qf-density15-ratio-proportional-partition","seed":202610152051,"semantic_template":"qf-density15-ratio-proportional-partition/[\"0/1\",\"0/1\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=1, b=1, same-unit nonnegative total T=0/1. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(0/1, 0/1, 0/1, 0/1, No, undefined)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (0/1, 0/1, 0/1, 0/1, No, undefined)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0052","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33a9a6d28f16daf498d7d100fc89b5d2bfa0e554a306a810c25078b61760a9d4","family_id":"qf-density15-ratio-proportional-partition","seed":202610152052,"semantic_template":"qf-density15-ratio-proportional-partition/[\"2/1\",\"6/1\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=1, b=3, same-unit nonnegative total T=8/1. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(2/1, 6/1, 0/1, 0/1, Yes, 1/3)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (2/1, 6/1, 0/1, 0/1, Yes, 1/3)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0053","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9f1acb6564e54d712d333fd3493bc43f10858274be49dd162ec3ae113330b43","family_id":"qf-density15-ratio-proportional-partition","seed":202610152053,"semantic_template":"qf-density15-ratio-proportional-partition/[\"3/2\",\"1/1\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=3, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(3/2, 1/1, 0/1, 0/1, Yes, 3/2)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (3/2, 1/1, 0/1, 0/1, Yes, 3/2)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0054","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38eb4b54bd5b5358909f420c49939ee25b0756bf5a0190710b576b8017c4bb69","family_id":"qf-density15-ratio-proportional-partition","seed":202610152054,"semantic_template":"qf-density15-ratio-proportional-partition/[\"5/3\",\"5/6\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=4, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(5/3, 5/6, 0/1, 0/1, Yes, 2/1)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (5/3, 5/6, 0/1, 0/1, Yes, 2/1)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0055","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bd361980a8aac86016c8729f493b74992daae830440413dfe5f9a550370ee62","family_id":"qf-density15-ratio-proportional-partition","seed":202610152055,"semantic_template":"qf-density15-ratio-proportional-partition/[\"25/14\",\"5/7\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=5, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(25/14, 5/7, 0/1, 0/1, Yes, 5/2)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (25/14, 5/7, 0/1, 0/1, Yes, 5/2)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0056","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ab953754c52d737c8699b4bc3a3d699d7f6fac932f38167c7f0a7699b3dbce1","family_id":"qf-density15-ratio-proportional-partition","seed":202610152056,"semantic_template":"qf-density15-ratio-proportional-partition/[\"15/8\",\"5/8\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=6, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(15/8, 5/8, 0/1, 0/1, Yes, 3/1)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (15/8, 5/8, 0/1, 0/1, Yes, 3/1)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0057","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"703245de1d76882062b8a5f9df0120c87ac891a60343786c32201e306222599b","family_id":"qf-density15-ratio-proportional-partition","seed":202610152057,"semantic_template":"qf-density15-ratio-proportional-partition/[\"35/18\",\"5/9\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=7, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(35/18, 5/9, 0/1, 0/1, Yes, 7/2)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (35/18, 5/9, 0/1, 0/1, Yes, 7/2)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0058","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f84329abfd06f4651cbea92dd5746b7622b300977499c2808cd7ceb04c863912","family_id":"qf-density15-ratio-proportional-partition","seed":202610152058,"semantic_template":"qf-density15-ratio-proportional-partition/[\"2/1\",\"1/2\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=8, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(2/1, 1/2, 0/1, 0/1, Yes, 4/1)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (2/1, 1/2, 0/1, 0/1, Yes, 4/1)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0059","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93a89cb7310f260894583ff5ca9cb3c1ec4dbc862b9d1e0f9979a1c6126e062a","family_id":"qf-density15-ratio-proportional-partition","seed":202610152059,"semantic_template":"qf-density15-ratio-proportional-partition/[\"45/22\",\"5/11\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=9, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(45/22, 5/11, 0/1, 0/1, Yes, 9/2)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (45/22, 5/11, 0/1, 0/1, Yes, 9/2)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0060","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18d23e234fc24cf0e501d73cfb55cb8c916521e7295e41ae9e6e2f2508b46436","family_id":"qf-density15-ratio-proportional-partition","seed":202610152060,"semantic_template":"qf-density15-ratio-proportional-partition/[\"25/12\",\"5/12\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=10, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(25/12, 5/12, 0/1, 0/1, Yes, 5/1)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (25/12, 5/12, 0/1, 0/1, Yes, 5/1)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0061","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fe6269db19dad72e27facb2c299b0a6bbdf90692686668eab877e09c8694e2d","family_id":"qf-density15-ratio-proportional-partition","seed":202610152061,"semantic_template":"qf-density15-ratio-proportional-partition/[\"55/26\",\"5/13\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=11, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(55/26, 5/13, 0/1, 0/1, Yes, 11/2)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (55/26, 5/13, 0/1, 0/1, Yes, 11/2)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0062","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24736bbb2e5982db86a5b938e9b143431bc1b0ccdd064464428b82048b1a4d91","family_id":"qf-density15-ratio-proportional-partition","seed":202610152062,"semantic_template":"qf-density15-ratio-proportional-partition/[\"15/7\",\"5/14\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=12, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(15/7, 5/14, 0/1, 0/1, Yes, 6/1)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (15/7, 5/14, 0/1, 0/1, Yes, 6/1)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0063","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c3426ccc160e2050bc0ca9f33a556eefe27245ead5ec93ebadc22d21fa4d63b","family_id":"qf-density15-ratio-proportional-partition","seed":202610152063,"semantic_template":"qf-density15-ratio-proportional-partition/[\"13/6\",\"1/3\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=13, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(13/6, 1/3, 0/1, 0/1, Yes, 13/2)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (13/6, 1/3, 0/1, 0/1, Yes, 13/2)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0064","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92f5f5b250fb0991916619a86fbe9230a508c3849ce99b9e66e7c404c17d4e5f","family_id":"qf-density15-ratio-proportional-partition","seed":202610152064,"semantic_template":"qf-density15-ratio-proportional-partition/[\"35/16\",\"5/16\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=14, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(35/16, 5/16, 0/1, 0/1, Yes, 7/1)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (35/16, 5/16, 0/1, 0/1, Yes, 7/1)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0065","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39031f8df4db9e16c6b05cc87c22bc9c89fbcf6cdb59134837b88a69dfa2d0df","family_id":"qf-density15-ratio-proportional-partition","seed":202610152065,"semantic_template":"qf-density15-ratio-proportional-partition/[\"75/34\",\"5/17\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=15, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(75/34, 5/17, 0/1, 0/1, Yes, 15/2)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (75/34, 5/17, 0/1, 0/1, Yes, 15/2)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0066","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac3156895b67df3c277634d1dd5185c50816c220e2d3444a3fecf604acc65a9a","family_id":"qf-density15-ratio-proportional-partition","seed":202610152066,"semantic_template":"qf-density15-ratio-proportional-partition/[\"20/9\",\"5/18\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=16, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(20/9, 5/18, 0/1, 0/1, Yes, 8/1)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (20/9, 5/18, 0/1, 0/1, Yes, 8/1)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0067","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19d5f43a5e7adb036e0d53ba54fd6b3dc7a955ab74c9a74261a1cb7fc6e2bbd8","family_id":"qf-density15-ratio-proportional-partition","seed":202610152067,"semantic_template":"qf-density15-ratio-proportional-partition/[\"85/38\",\"5/19\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=17, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(85/38, 5/19, 0/1, 0/1, Yes, 17/2)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (85/38, 5/19, 0/1, 0/1, Yes, 17/2)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0068","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e235dbd0e6b85a5abba57b1752829e895a3ba4f8832768320a41098250ce1f9d","family_id":"qf-density15-ratio-proportional-partition","seed":202610152068,"semantic_template":"qf-density15-ratio-proportional-partition/[\"9/4\",\"1/4\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=18, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(9/4, 1/4, 0/1, 0/1, Yes, 9/1)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (9/4, 1/4, 0/1, 0/1, Yes, 9/1)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0069","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7983dfb34d55482864cc56c1d1c2f79ce6a09914ed4d146dc9cb5cb58fead42","family_id":"qf-density15-ratio-proportional-partition","seed":202610152069,"semantic_template":"qf-density15-ratio-proportional-partition/[\"95/42\",\"5/21\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=19, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(95/42, 5/21, 0/1, 0/1, Yes, 19/2)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (95/42, 5/21, 0/1, 0/1, Yes, 19/2)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0070","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53b7e8b52b16c3a33596944587715b8b51830680c8d054c2abe24e4923454f11","family_id":"qf-density15-ratio-proportional-partition","seed":202610152070,"semantic_template":"qf-density15-ratio-proportional-partition/[\"25/11\",\"5/22\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=20, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(25/11, 5/22, 0/1, 0/1, Yes, 10/1)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (25/11, 5/22, 0/1, 0/1, Yes, 10/1)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0071","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f067989aa4458a43ac34c5eb579ee6face2d316a97861339e2332cf92067085","family_id":"qf-density15-ratio-proportional-partition","seed":202610152071,"semantic_template":"qf-density15-ratio-proportional-partition/[\"105/46\",\"5/23\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=21, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(105/46, 5/23, 0/1, 0/1, Yes, 21/2)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (105/46, 5/23, 0/1, 0/1, Yes, 21/2)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0072","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"564db4a559e0e1c85231bf992e6483dfe0e557e1083ceaf529973f9ae507dd5a","family_id":"qf-density15-ratio-proportional-partition","seed":202610152072,"semantic_template":"qf-density15-ratio-proportional-partition/[\"55/24\",\"5/24\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=22, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(55/24, 5/24, 0/1, 0/1, Yes, 11/1)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (55/24, 5/24, 0/1, 0/1, Yes, 11/1)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0073","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec26fb565de7911534a7b0095955c6be434410fa5cd8eef7ebe34c5aa57099e9","family_id":"qf-density15-ratio-proportional-partition","seed":202610152073,"semantic_template":"qf-density15-ratio-proportional-partition/[\"23/10\",\"1/5\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=23, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(23/10, 1/5, 0/1, 0/1, Yes, 23/2)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (23/10, 1/5, 0/1, 0/1, Yes, 23/2)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0074","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f37a9102c80bbeae7a6e6b4022f12e177b07dea383c2a6d3697d3bfea8bd789","family_id":"qf-density15-ratio-proportional-partition","seed":202610152074,"semantic_template":"qf-density15-ratio-proportional-partition/[\"30/13\",\"5/26\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=24, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(30/13, 5/26, 0/1, 0/1, Yes, 12/1)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (30/13, 5/26, 0/1, 0/1, Yes, 12/1)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0075","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e56e6814ff4c7c350998046bf433eb3ad9228a92b4aa62506a2704e366e4fac6","family_id":"qf-density15-ratio-proportional-partition","seed":202610152075,"semantic_template":"qf-density15-ratio-proportional-partition/[\"125/54\",\"5/27\"]"},"fields":{"title":"Same-Unit Weighted Partition and Actual Ratio Guard","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=25, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).","answer":{"kind":"expression","value":"(125/54, 5/27, 0/1, 0/1, Yes, 25/2)","explanation":"Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (125/54, 5/27, 0/1, 0/1, Yes, 25/2)."},"skills":["Same-Unit Weighted Partition and Actual Ratio Guard"],"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-20261011-advance-20261015-r02-ratio-proportion-0076","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b419a79b7aec8e728e4e9d54daa958abc1525ee3323f6915a6895ae8b2cf800d","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152076,"semantic_template":"qf-density15-ratio-missing-component-guard/[false,\"undefined\",\"0/1\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=0, b=0, d=0, x=0/1. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(No, undefined, 0/1, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (No, undefined, 0/1, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0077","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b934378c75637b150c2fe7a6f32aca68098da9e301faab1a717cf3938ecbeeda","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152077,"semantic_template":"qf-density15-ratio-missing-component-guard/[false,\"undefined\",\"2/1\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=1, b=0, d=2, x=3/1. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(No, undefined, 2/1, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (No, undefined, 2/1, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0078","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a687ff4466968322ec5b97ebc2b2b063ee161a1063c35654d3562a0c236da748","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152078,"semantic_template":"qf-density15-ratio-missing-component-guard/[false,\"undefined\",\"-1/1\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=2, b=3, d=0, x=1/3. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(No, undefined, -1/1, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (No, undefined, -1/1, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0079","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c88c51ff9402297390086237797e298a4de90d758f7a2d386f38a24a1a4e0e6c","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152079,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"3/2\",\"0/1\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=1, b=2, d=3, x=3/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, 3/2, 0/1, Yes)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 3/2, 0/1, Yes)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0080","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52ee416b01c65ca24e36049ee39b201db1779dfbc8a326334364d78835487771","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152080,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"-16/3\",\"-47/2\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=-8, b=3, d=2, x=5/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, -16/3, -47/2, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, -16/3, -47/2, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0081","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b28df0790403170f6076c674840cb720772d74cd7663efc6e06d5dd8df046d4e","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152081,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"-14/3\",\"-23/1\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=-7, b=3, d=2, x=6/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, -14/3, -23/1, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, -14/3, -23/1, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0082","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a95842ec0fff2454a8d448230a20c71d35294ae3f5caf1ef24335f537c6e95ab","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152082,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"-4/1\",\"-45/2\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=-6, b=3, d=2, x=7/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, -4/1, -45/2, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, -4/1, -45/2, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0083","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e37e904be06fc26b72d6f732caad9dabca9566dc43045018043cbbe938fccedc","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152083,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"-10/3\",\"-22/1\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=-5, b=3, d=2, x=8/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, -10/3, -22/1, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, -10/3, -22/1, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0084","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebcad533214b35a6d23af60e3b6381a0bd6da9b7389571e44592045a093273eb","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152084,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"-8/3\",\"-43/2\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=-4, b=3, d=2, x=9/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, -8/3, -43/2, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, -8/3, -43/2, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0085","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0201b45525a2ed8666572831313c7eda66078414064f308fd2a7b0f05fd2ed2","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152085,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"-2/1\",\"-21/1\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=-3, b=3, d=2, x=10/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, -2/1, -21/1, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, -2/1, -21/1, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0086","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e05c4f68f57a4f2d229decfce6b3c81c293a7b4411894fa015f800f7411e853","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152086,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"-4/3\",\"-41/2\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=-2, b=3, d=2, x=11/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, -4/3, -41/2, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, -4/3, -41/2, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0087","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03e4dbac99e8ff6eb974a21817314ce5d3641cc27c3f96fa34213802c9b43842","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152087,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"-2/3\",\"-20/1\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=-1, b=3, d=2, x=12/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, -2/3, -20/1, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, -2/3, -20/1, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0088","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dad05949f43135f154d0600922634923c4461fc604cfdc6f647bac484e180f56","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152088,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"0/1\",\"-39/2\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=0, b=3, d=2, x=13/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, 0/1, -39/2, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 0/1, -39/2, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0089","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fa8511e53195ef4bf86a0033d418295453bfb7e41159caa0d3d989d3ae599dc","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152089,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"2/3\",\"-19/1\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=1, b=3, d=2, x=14/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, 2/3, -19/1, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 2/3, -19/1, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0090","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bf015d5b3536fe12e100e563283b2571e1e45d6fa9830942dedfebfa8215676","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152090,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"4/3\",\"-37/2\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=2, b=3, d=2, x=15/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, 4/3, -37/2, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 4/3, -37/2, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0091","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8466b11397b60736ed98b2079da1e1e6b2d9e3e77401d784702ec82d6bdc102","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152091,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"2/1\",\"-18/1\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=3, b=3, d=2, x=16/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, 2/1, -18/1, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 2/1, -18/1, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0092","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8f97918c063d9bcc25c22bf693999d9bbbf4e44c6698a339b4a18c8d42830c4","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152092,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"8/3\",\"-35/2\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=4, b=3, d=2, x=17/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, 8/3, -35/2, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 8/3, -35/2, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0093","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55e5647f05ec3934961f49cef4cdf92c16439bb8fbb39566006c1fae47e9af07","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152093,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"10/3\",\"-17/1\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=5, b=3, d=2, x=18/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, 10/3, -17/1, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 10/3, -17/1, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0094","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acc9382d9f69551191d8ede7b9ad2f649345dafc7dcf9e18936e69b73661ff0d","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152094,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"4/1\",\"-33/2\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=6, b=3, d=2, x=19/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, 4/1, -33/2, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 4/1, -33/2, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0095","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9b02bed7ad136078770110e5b16d0f57494afe2d161bae2bca380b571c64739","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152095,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"14/3\",\"-16/1\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=7, b=3, d=2, x=20/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, 14/3, -16/1, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 14/3, -16/1, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0096","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcff836bfc3b7e8575826770ab87fdbc3a4de06a3fd5076e9211e06114676bd8","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152096,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"16/3\",\"-31/2\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=8, b=3, d=2, x=21/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, 16/3, -31/2, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 16/3, -31/2, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0097","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffad04184c7d12a4148d832e306fba5760ff1fa951831513c763fd283d3cede7","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152097,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"6/1\",\"-15/1\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=9, b=3, d=2, x=22/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, 6/1, -15/1, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 6/1, -15/1, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0098","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54576090656f286d5c7f75cf1ef1d5040b9fd5d5908426c3037847ed6f2ba8ee","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152098,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"20/3\",\"-29/2\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=10, b=3, d=2, x=23/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, 20/3, -29/2, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 20/3, -29/2, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0099","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a282b984605d67ceee7fe995bfc1821b13d3e6ce5cbb6cae9af8a90d0216264b","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152099,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"22/3\",\"-14/1\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=11, b=3, d=2, x=24/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, 22/3, -14/1, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 22/3, -14/1, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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-20261011-advance-20261015-r02-ratio-proportion-0100","batch_id":"batch-20261011-advance-20261015-r02-ratio-proportion-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bb846364bb2316ad0130dd6d63777e7d44aad3ae4a63040f75c155a0c9254ea","family_id":"qf-density15-ratio-missing-component-guard","seed":202610152100,"semantic_template":"qf-density15-ratio-missing-component-guard/[true,\"8/1\",\"-27/2\"]"},"fields":{"title":"Candidate Proportion Guards and Cross-Product Defect","stem":"Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=12, b=3, d=2, x=25/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).","answer":{"kind":"expression","value":"(Yes, 8/1, -27/2, No)","explanation":"Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 8/1, -27/2, No)."},"skills":["Candidate Proportion Guards and Cross-Product 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."}}