{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-terminating-decimal-fraction-0001","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfb369c07e6ef02143d18dc85a4b2e0f14eecd5c9ab77ada1fae62d89903db60","family_id":"qf-real-terminating-decimal-fraction","seed":20260924,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"0.5\",\"operation\":\"decimal_to_fraction\",\"scale\":1}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 0.5 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"1/2","explanation":"0.5 is converted to an integer numerator over 10^1 and reduced by the gcd, giving 1/2."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0002","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a48f89213fdd455d8050ac8cf151377a98a87213822d19c7f158ed0e4fd25027","family_id":"qf-real-terminating-decimal-fraction","seed":20260925,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"1.25\",\"operation\":\"decimal_to_fraction\",\"scale\":2}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 1.25 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"5/4","explanation":"1.25 is converted to an integer numerator over 10^2 and reduced by the gcd, giving 5/4."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0003","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4e12c62122df14c501fee5d2584f5153542ac247276c083962354dc4e53a636","family_id":"qf-real-terminating-decimal-fraction","seed":20260926,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"-2.4\",\"operation\":\"decimal_to_fraction\",\"scale\":1}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string -2.4 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"-12/5","explanation":"-2.4 is converted to an integer numerator over 10^1 and reduced by the gcd, giving -12/5."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0004","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a364da388293515b06270e20fa65841b58b01880ce013fd20affba7ad55665d","family_id":"qf-real-terminating-decimal-fraction","seed":20260927,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"3.125\",\"operation\":\"decimal_to_fraction\",\"scale\":3}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 3.125 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"25/8","explanation":"3.125 is converted to an integer numerator over 10^3 and reduced by the gcd, giving 25/8."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0005","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1aa0aa5cbcd9c54776bed19447e253cbf5f2e803069ca39c0642d61f061419be","family_id":"qf-real-terminating-decimal-fraction","seed":20260928,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"0.04\",\"operation\":\"decimal_to_fraction\",\"scale\":2}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 0.04 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"1/25","explanation":"0.04 is converted to an integer numerator over 10^2 and reduced by the gcd, giving 1/25."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0006","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7178ee2eaf952c3a0401f5ff3799b92cc1553669932eb7ac843d346f6c9796e1","family_id":"qf-real-terminating-decimal-fraction","seed":20260929,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"-0.75\",\"operation\":\"decimal_to_fraction\",\"scale\":2}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string -0.75 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"-3/4","explanation":"-0.75 is converted to an integer numerator over 10^2 and reduced by the gcd, giving -3/4."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0007","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f51826b0263a805cd4a8be3792469d86189e5d17e8ce42848aeb03bd7279ef90","family_id":"qf-real-terminating-decimal-fraction","seed":20260930,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"12.5\",\"operation\":\"decimal_to_fraction\",\"scale\":1}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 12.5 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"25/2","explanation":"12.5 is converted to an integer numerator over 10^1 and reduced by the gcd, giving 25/2."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0008","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fed0df805a0d49dbab837ec04c87c6551b37110fdb0cd23a39fb67c659979358","family_id":"qf-real-terminating-decimal-fraction","seed":20260931,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"0.125\",\"operation\":\"decimal_to_fraction\",\"scale\":3}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 0.125 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"1/8","explanation":"0.125 is converted to an integer numerator over 10^3 and reduced by the gcd, giving 1/8."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0009","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47b236311d66531edead11d27731e89ab16b82195c0eb58db7a273d69e81ab1d","family_id":"qf-real-terminating-decimal-fraction","seed":20260932,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"2.05\",\"operation\":\"decimal_to_fraction\",\"scale\":2}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 2.05 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"41/20","explanation":"2.05 is converted to an integer numerator over 10^2 and reduced by the gcd, giving 41/20."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0010","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"728e9b2f6451d55c44f5a900b91a7d4191317c4dc022ee3375052563dda4fb74","family_id":"qf-real-terminating-decimal-fraction","seed":20260933,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"-1.2\",\"operation\":\"decimal_to_fraction\",\"scale\":1}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string -1.2 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"-6/5","explanation":"-1.2 is converted to an integer numerator over 10^1 and reduced by the gcd, giving -6/5."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0011","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9691b3a391a4dfcfa6f66d8fc66527b51f81c6ab369146e8015fc4f0cc5cd18","family_id":"qf-real-terminating-decimal-fraction","seed":20260934,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"4.375\",\"operation\":\"decimal_to_fraction\",\"scale\":3}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 4.375 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"35/8","explanation":"4.375 is converted to an integer numerator over 10^3 and reduced by the gcd, giving 35/8."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0012","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d07d51e6fb1317d528db76dceaf53e5ac22ac7fd87d7f002486c12d6e753a0ea","family_id":"qf-real-terminating-decimal-fraction","seed":20260935,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"0.625\",\"operation\":\"decimal_to_fraction\",\"scale\":3}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 0.625 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"5/8","explanation":"0.625 is converted to an integer numerator over 10^3 and reduced by the gcd, giving 5/8."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0013","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c07ccc82d8340bb575eb4bed1db30a88cf4c1fdfd4cccc6839666b78096fc7c","family_id":"qf-real-terminating-decimal-fraction","seed":20260936,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"7.01\",\"operation\":\"decimal_to_fraction\",\"scale\":2}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 7.01 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"701/100","explanation":"7.01 is converted to an integer numerator over 10^2 and reduced by the gcd, giving 701/100."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0014","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1aed4a3eca9ecbd478e73bf7f225738963ccc5961a9ff0967f9f02d6a489811a","family_id":"qf-real-terminating-decimal-fraction","seed":20260937,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"-3.5\",\"operation\":\"decimal_to_fraction\",\"scale\":1}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string -3.5 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"-7/2","explanation":"-3.5 is converted to an integer numerator over 10^1 and reduced by the gcd, giving -7/2."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0015","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0153c1e12e7139e18c914443969f09c7600dfd010012013a6a3d518b271e1955","family_id":"qf-real-terminating-decimal-fraction","seed":20260938,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"8.2\",\"operation\":\"decimal_to_fraction\",\"scale\":1}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 8.2 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"41/5","explanation":"8.2 is converted to an integer numerator over 10^1 and reduced by the gcd, giving 41/5."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0016","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9633be8effeb5e3b0d2f13f8dd05f78a2d7fe953b123b8ef91cd0d09e060ab6a","family_id":"qf-real-terminating-decimal-fraction","seed":20260939,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"0.09\",\"operation\":\"decimal_to_fraction\",\"scale\":2}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 0.09 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"9/100","explanation":"0.09 is converted to an integer numerator over 10^2 and reduced by the gcd, giving 9/100."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0017","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e20a26ca039b596cf2a9f4ada6762b8b83a0b09326778ff640b495c0e99b890","family_id":"qf-real-terminating-decimal-fraction","seed":20260940,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"-0.01\",\"operation\":\"decimal_to_fraction\",\"scale\":2}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string -0.01 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"-1/100","explanation":"-0.01 is converted to an integer numerator over 10^2 and reduced by the gcd, giving -1/100."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0018","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"218e6e21264adcecac9c41dc42a0ab64389ea9eb9bceb994df4c439808b216d1","family_id":"qf-real-terminating-decimal-fraction","seed":20260941,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"15.625\",\"operation\":\"decimal_to_fraction\",\"scale\":3}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 15.625 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"125/8","explanation":"15.625 is converted to an integer numerator over 10^3 and reduced by the gcd, giving 125/8."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0019","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7951d6489238d663467376bae9e6271b79a7454e5a60b53c078713b435f424f0","family_id":"qf-real-terminating-decimal-fraction","seed":20260942,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"6.4\",\"operation\":\"decimal_to_fraction\",\"scale\":1}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 6.4 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"32/5","explanation":"6.4 is converted to an integer numerator over 10^1 and reduced by the gcd, giving 32/5."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0020","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00dcc1b172e3c2d68bd8529237fdca775f0f21291c36e47ed5dffb942e513ce8","family_id":"qf-real-terminating-decimal-fraction","seed":20260943,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"-9.08\",\"operation\":\"decimal_to_fraction\",\"scale\":2}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string -9.08 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"-227/25","explanation":"-9.08 is converted to an integer numerator over 10^2 and reduced by the gcd, giving -227/25."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0021","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eef5074a9b20e5e81c7a96613c247f2b7df1382dc0881c2d5902e096ef0f3917","family_id":"qf-real-terminating-decimal-fraction","seed":20260944,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"0.2\",\"operation\":\"decimal_to_fraction\",\"scale\":1}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 0.2 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"1/5","explanation":"0.2 is converted to an integer numerator over 10^1 and reduced by the gcd, giving 1/5."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0022","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30e62795d5ac9bb99c7f4a3714a8a9356c7e6796777fa19c989b8921f1fd6dec","family_id":"qf-real-terminating-decimal-fraction","seed":20260945,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"11.25\",\"operation\":\"decimal_to_fraction\",\"scale\":2}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 11.25 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"45/4","explanation":"11.25 is converted to an integer numerator over 10^2 and reduced by the gcd, giving 45/4."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0023","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f26ea07560a301f29c3b07920db1d104a8c100615122c15233642c64cbf544b","family_id":"qf-real-terminating-decimal-fraction","seed":20260946,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"-4.125\",\"operation\":\"decimal_to_fraction\",\"scale\":3}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string -4.125 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"-33/8","explanation":"-4.125 is converted to an integer numerator over 10^3 and reduced by the gcd, giving -33/8."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0024","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"716a87c8cb9944492abe0a85734c54b26b082f8d810dfd6a8650750117ebc7f0","family_id":"qf-real-terminating-decimal-fraction","seed":20260947,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"2.75\",\"operation\":\"decimal_to_fraction\",\"scale\":2}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 2.75 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"11/4","explanation":"2.75 is converted to an integer numerator over 10^2 and reduced by the gcd, giving 11/4."},"skills":["terminating decimals","gcd"],"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-20260923-real-terminating-decimal-fraction-0025","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"974335b0c74db3ba10ebb0d606f0f096b7996d7b14ea3a98a87fa9fdc426c000","family_id":"qf-real-terminating-decimal-fraction","seed":20260948,"semantic_template":"qf-real-terminating-decimal-fraction/{\"decimal\":\"0.001\",\"operation\":\"decimal_to_fraction\",\"scale\":3}"},"fields":{"title":"Exact Fraction for a Terminating Decimal","stem":"Convert the explicit decimal string 0.001 exactly to a lowest-terms fraction with a positive denominator; do not use floating-point approximation.","answer":{"kind":"short_text","value":"1/1000","explanation":"0.001 is converted to an integer numerator over 10^3 and reduced by the gcd, giving 1/1000."},"skills":["terminating decimals","gcd"],"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-20260923-real-rational-order-0001","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4110f227bc6ba4bfe12110851f4ef346a3172cf8e7dcf624c74d69c0c33f713e","family_id":"qf-real-rational-order","seed":20261024,"semantic_template":"qf-real-rational-order/{\"left\":[-12,1],\"operation\":\"cross_multiplication_order\",\"right\":[-6,1]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=-12/1 and b=-6/1. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":"<","explanation":"-12×1 < -6×1, so -12/1 < -6/1."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0002","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"794f0553a6cbef5d7b269c667a8e197ac41657697bf24802e45fa5a52432ffad","family_id":"qf-real-rational-order","seed":20261025,"semantic_template":"qf-real-rational-order/{\"left\":[-11,2],\"operation\":\"cross_multiplication_order\",\"right\":[-5,2]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=-11/2 and b=-5/2. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":"<","explanation":"-11×2 < -5×2, so -11/2 < -5/2."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0003","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bf541da2c7c657e8a8f200614893990d906623c6b2c39cab514a3bbf7388d3a","family_id":"qf-real-rational-order","seed":20261026,"semantic_template":"qf-real-rational-order/{\"left\":[-10,3],\"operation\":\"cross_multiplication_order\",\"right\":[-4,3]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=-10/3 and b=-4/3. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":"<","explanation":"-10×3 < -4×3, so -10/3 < -4/3."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0004","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d41d8caa7fc05f68962d9205ba5d98341187847b0acda017e8a62c07cfc84370","family_id":"qf-real-rational-order","seed":20261027,"semantic_template":"qf-real-rational-order/{\"left\":[-9,4],\"operation\":\"cross_multiplication_order\",\"right\":[-3,4]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=-9/4 and b=-3/4. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":"<","explanation":"-9×4 < -3×4, so -9/4 < -3/4."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0005","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b65ab0df234219cb04aa43611d0db9ed50c4f703871333e0cfbfdee140057699","family_id":"qf-real-rational-order","seed":20261028,"semantic_template":"qf-real-rational-order/{\"left\":[-8,5],\"operation\":\"cross_multiplication_order\",\"right\":[-2,5]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=-8/5 and b=-2/5. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":"<","explanation":"-8×5 < -2×5, so -8/5 < -2/5."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0006","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f5add97e4c194f71dc80c3fe4d1b709ee92e4072cb82c636cd508e55eccaaff","family_id":"qf-real-rational-order","seed":20261029,"semantic_template":"qf-real-rational-order/{\"left\":[-7,6],\"operation\":\"cross_multiplication_order\",\"right\":[-1,1]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=-7/6 and b=-1/1. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":"<","explanation":"-7×1 < -1×6, so -7/6 < -1/1."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0007","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acea350170ab54cb1c767813dc5f456eea3d7e90b60bbd97caf71038fb2a94a3","family_id":"qf-real-rational-order","seed":20261030,"semantic_template":"qf-real-rational-order/{\"left\":[-6,7],\"operation\":\"cross_multiplication_order\",\"right\":[0,1]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=-6/7 and b=0/1. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":"<","explanation":"-6×1 < 0×7, so -6/7 < 0/1."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0008","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3dce0dc58e7a0279107e6ab844af53950dc61f8d69c074214f62a829bc26e55","family_id":"qf-real-rational-order","seed":20261031,"semantic_template":"qf-real-rational-order/{\"left\":[-5,1],\"operation\":\"cross_multiplication_order\",\"right\":[1,3]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=-5/1 and b=1/3. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":"<","explanation":"-5×3 < 1×1, so -5/1 < 1/3."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0009","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9183cce176a55042818d9126b0d1ef488f201d31c1d4326794a965545dde881","family_id":"qf-real-rational-order","seed":20261032,"semantic_template":"qf-real-rational-order/{\"left\":[-2,1],\"operation\":\"cross_multiplication_order\",\"right\":[1,2]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=-2/1 and b=1/2. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":"<","explanation":"-2×2 < 1×1, so -2/1 < 1/2."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0010","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19574c4319b0dc8c66ab8a6bfe080aab466c0d8ca60296bb322aea0dbdf51c31","family_id":"qf-real-rational-order","seed":20261033,"semantic_template":"qf-real-rational-order/{\"left\":[-1,1],\"operation\":\"cross_multiplication_order\",\"right\":[3,5]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=-1/1 and b=3/5. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":"<","explanation":"-1×5 < 3×1, so -1/1 < 3/5."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0011","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"054e734dfe03c772fd42414f7d855ceabbce1f118e964579a44ff8977622bd5a","family_id":"qf-real-rational-order","seed":20261034,"semantic_template":"qf-real-rational-order/{\"left\":[-1,2],\"operation\":\"cross_multiplication_order\",\"right\":[4,1]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=-1/2 and b=4/1. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":"<","explanation":"-1×1 < 4×2, so -1/2 < 4/1."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0012","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdaabb306f4822fa3b77d5142600c7a29ca2ecff4f56310fb44bc54881f9ac63","family_id":"qf-real-rational-order","seed":20261035,"semantic_template":"qf-real-rational-order/{\"left\":[-1,5],\"operation\":\"cross_multiplication_order\",\"right\":[5,2]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=-1/5 and b=5/2. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":"<","explanation":"-1×2 < 5×5, so -1/5 < 5/2."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0013","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3936459f3f36155f23c01689367b083aa98df6d141c97117f4b5f73f394abfb","family_id":"qf-real-rational-order","seed":20261036,"semantic_template":"qf-real-rational-order/{\"left\":[0,1],\"operation\":\"cross_multiplication_order\",\"right\":[2,1]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=0/1 and b=2/1. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":"<","explanation":"0×1 < 2×1, so 0/1 < 2/1."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0014","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a6ce0beae85336db0164d3f445037312d63e5cf23e6cad68d265d7802247cdc","family_id":"qf-real-rational-order","seed":20261037,"semantic_template":"qf-real-rational-order/{\"left\":[1,7],\"operation\":\"cross_multiplication_order\",\"right\":[-3,2]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=1/7 and b=-3/2. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":">","explanation":"1×2 > -3×7, so 1/7 > -3/2."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0015","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc2c39d63a7da35bc33d35bd38e19534f0715c977cb1e50f3bd432304dab3ebd","family_id":"qf-real-rational-order","seed":20261038,"semantic_template":"qf-real-rational-order/{\"left\":[2,1],\"operation\":\"cross_multiplication_order\",\"right\":[-1,1]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=2/1 and b=-1/1. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":">","explanation":"2×1 > -1×1, so 2/1 > -1/1."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0016","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12380d5c23f4ace8942dcd417e12747df2511311112437776597792c1d19d087","family_id":"qf-real-rational-order","seed":20261039,"semantic_template":"qf-real-rational-order/{\"left\":[3,2],\"operation\":\"cross_multiplication_order\",\"right\":[-4,1]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=3/2 and b=-4/1. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":">","explanation":"3×1 > -4×2, so 3/2 > -4/1."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0017","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b70604136ac1cdcedd98f2e587a47b6cfad38231bef3b443a58b686a8729f03c","family_id":"qf-real-rational-order","seed":20261040,"semantic_template":"qf-real-rational-order/{\"left\":[4,3],\"operation\":\"cross_multiplication_order\",\"right\":[-3,2]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=4/3 and b=-3/2. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":">","explanation":"4×2 > -3×3, so 4/3 > -3/2."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0018","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ad3eb1f6d7114aa31db158cbf4f1e7970b8b4d258393b4f018a954a634e44e8","family_id":"qf-real-rational-order","seed":20261041,"semantic_template":"qf-real-rational-order/{\"left\":[5,4],\"operation\":\"cross_multiplication_order\",\"right\":[-2,3]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=5/4 and b=-2/3. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":">","explanation":"5×3 > -2×4, so 5/4 > -2/3."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0019","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3688a74927b08411953a90ed5aac8a479565a1b1f65a4b2fa8d2898e6c5f434","family_id":"qf-real-rational-order","seed":20261042,"semantic_template":"qf-real-rational-order/{\"left\":[6,5],\"operation\":\"cross_multiplication_order\",\"right\":[-1,4]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=6/5 and b=-1/4. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":">","explanation":"6×4 > -1×5, so 6/5 > -1/4."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0020","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac06f6f098c47caeb786c0c442a3562689be68da7ae1a7c08ca50fadb05e8eb4","family_id":"qf-real-rational-order","seed":20261043,"semantic_template":"qf-real-rational-order/{\"left\":[7,6],\"operation\":\"cross_multiplication_order\",\"right\":[0,1]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=7/6 and b=0/1. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":">","explanation":"7×1 > 0×6, so 7/6 > 0/1."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0021","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6dbdc47345be33b642d10d514166e7f0c7ddbcc4b229f35a48bf0aef015724a","family_id":"qf-real-rational-order","seed":20261044,"semantic_template":"qf-real-rational-order/{\"left\":[8,7],\"operation\":\"cross_multiplication_order\",\"right\":[1,1]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=8/7 and b=1/1. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":">","explanation":"8×1 > 1×7, so 8/7 > 1/1."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0022","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30d7fea7ed4ed9024fc6406de6f810863eca80a26e894783568af214c071c030","family_id":"qf-real-rational-order","seed":20261045,"semantic_template":"qf-real-rational-order/{\"left\":[9,1],\"operation\":\"cross_multiplication_order\",\"right\":[1,1]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=9/1 and b=1/1. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":">","explanation":"9×1 > 1×1, so 9/1 > 1/1."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0023","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cbb55833362410a83cfb4dd8bf687780db92004b443ebaf41d6109108295e8b","family_id":"qf-real-rational-order","seed":20261046,"semantic_template":"qf-real-rational-order/{\"left\":[5,1],\"operation\":\"cross_multiplication_order\",\"right\":[1,1]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=5/1 and b=1/1. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":">","explanation":"5×1 > 1×1, so 5/1 > 1/1."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0024","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3bcf827a0b31c62ae89df0b2e6eae80cdd01a814dbc756682c4251f47898685","family_id":"qf-real-rational-order","seed":20261047,"semantic_template":"qf-real-rational-order/{\"left\":[11,3],\"operation\":\"cross_multiplication_order\",\"right\":[1,1]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=11/3 and b=1/1. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":">","explanation":"11×1 > 1×3, so 11/3 > 1/1."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-order-0025","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3eaf7cc746f989b1fbceba84334137d99ec89b0e17d3885e0f3592124f3493b1","family_id":"qf-real-rational-order","seed":20261048,"semantic_template":"qf-real-rational-order/{\"left\":[3,1],\"operation\":\"cross_multiplication_order\",\"right\":[1,1]}"},"fields":{"title":"Exact Ordering of Rational Points","stem":"The rational points are a=3/1 and b=1/1. Use cross multiplication to determine whether a<b, a=b, or a>b.","answer":{"kind":"short_text","value":">","explanation":"3×1 > 1×1, so 3/1 > 1/1."},"skills":["rational numbers","cross multiplication"],"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-20260923-real-rational-closed-interval-0001","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54e1ca56f4e77498909973fb2e07ec6c3c6116fec85c9c9bcd24402156ca13fd","family_id":"qf-real-rational-closed-interval","seed":20261124,"semantic_template":"qf-real-rational-closed-interval/{\"a\":-12,\"b\":-11,\"operation\":\"closed_interval\",\"point\":[-12,1]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [-12, -11] and rational point -12/1 are explicit. Does -12/1 belong to this closed interval?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"Exact cross multiplication confirms -12≤-12/1≤-11; equality with an endpoint is included in a closed interval."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0002","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c5d5a7e536a3e6260e5efa88f8bd71a28aadc73b3e7891d1da34b6ba1cdeda5","family_id":"qf-real-rational-closed-interval","seed":20261125,"semantic_template":"qf-real-rational-closed-interval/{\"a\":-11,\"b\":-9,\"operation\":\"closed_interval\",\"point\":[-5,1]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [-11, -9] and rational point -5/1 are explicit. Does -5/1 belong to this closed interval?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"Exact cross multiplication shows that -5/1 lies outside [-11, -9]."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0003","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c6e1fb804e9843e733b9b30c646ecd2afb5eff70b0f7e50c605d722de189199","family_id":"qf-real-rational-closed-interval","seed":20261126,"semantic_template":"qf-real-rational-closed-interval/{\"a\":-10,\"b\":-7,\"operation\":\"closed_interval\",\"point\":[-8,3]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [-10, -7] and rational point -8/3 are explicit. Does -8/3 belong to this closed interval?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"Exact cross multiplication shows that -8/3 lies outside [-10, -7]."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0004","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3be315611bf0faf9116d96f293394df8e18640ff28a82637a0a03044bb50dec6","family_id":"qf-real-rational-closed-interval","seed":20261127,"semantic_template":"qf-real-rational-closed-interval/{\"a\":-9,\"b\":-5,\"operation\":\"closed_interval\",\"point\":[-3,2]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [-9, -5] and rational point -3/2 are explicit. Does -3/2 belong to this closed interval?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"Exact cross multiplication shows that -3/2 lies outside [-9, -5]."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0005","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7339ad6d73f8ddd0b1eef2ef13ff13257ca66feba00cb44743393039e0f85ad7","family_id":"qf-real-rational-closed-interval","seed":20261128,"semantic_template":"qf-real-rational-closed-interval/{\"a\":-8,\"b\":-3,\"operation\":\"closed_interval\",\"point\":[-4,1]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [-8, -3] and rational point -4/1 are explicit. Does -4/1 belong to this closed interval?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"Exact cross multiplication confirms -8≤-4/1≤-3; equality with an endpoint is included in a closed interval."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0006","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b23d210ffdf107558325bfead9a4c97458a41b5d51a4fe2385623d011cbd787","family_id":"qf-real-rational-closed-interval","seed":20261129,"semantic_template":"qf-real-rational-closed-interval/{\"a\":-7,\"b\":-1,\"operation\":\"closed_interval\",\"point\":[-1,1]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [-7, -1] and rational point -1/1 are explicit. Does -1/1 belong to this closed interval?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"Exact cross multiplication confirms -7≤-1/1≤-1; equality with an endpoint is included in a closed interval."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0007","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5aef5a14332267e300f5eace286f943bc3beff2e5639b52b77002eb7a67eb37","family_id":"qf-real-rational-closed-interval","seed":20261130,"semantic_template":"qf-real-rational-closed-interval/{\"a\":-6,\"b\":-5,\"operation\":\"closed_interval\",\"point\":[-2,1]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [-6, -5] and rational point -2/1 are explicit. Does -2/1 belong to this closed interval?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"Exact cross multiplication shows that -2/1 lies outside [-6, -5]."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0008","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7d0cd52fbf9dd21f6e712c92c00b1cb75df9e03cc4d1bcde1013398aa833f6a","family_id":"qf-real-rational-closed-interval","seed":20261131,"semantic_template":"qf-real-rational-closed-interval/{\"a\":-5,\"b\":-3,\"operation\":\"closed_interval\",\"point\":[-1,1]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [-5, -3] and rational point -1/1 are explicit. Does -1/1 belong to this closed interval?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"Exact cross multiplication shows that -1/1 lies outside [-5, -3]."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0009","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3759793f7b43a4b83f9e6d07d3848ac23adea85f9f314665a0806783a86eceb","family_id":"qf-real-rational-closed-interval","seed":20261132,"semantic_template":"qf-real-rational-closed-interval/{\"a\":-4,\"b\":-1,\"operation\":\"closed_interval\",\"point\":[-4,1]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [-4, -1] and rational point -4/1 are explicit. Does -4/1 belong to this closed interval?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"Exact cross multiplication confirms -4≤-4/1≤-1; equality with an endpoint is included in a closed interval."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0010","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"137bb7643c6dfe5ba329c12418e9973e956d8dc94a744138374964001ba2a359","family_id":"qf-real-rational-closed-interval","seed":20261133,"semantic_template":"qf-real-rational-closed-interval/{\"a\":-3,\"b\":1,\"operation\":\"closed_interval\",\"point\":[1,2]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [-3, 1] and rational point 1/2 are explicit. Does 1/2 belong to this closed interval?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"Exact cross multiplication confirms -3≤1/2≤1; equality with an endpoint is included in a closed interval."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0011","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b0b1c8f0b0e6123bedba465f748a0e54d27dc855ca381a01b8921491fe5736c","family_id":"qf-real-rational-closed-interval","seed":20261134,"semantic_template":"qf-real-rational-closed-interval/{\"a\":-2,\"b\":3,\"operation\":\"closed_interval\",\"point\":[2,3]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [-2, 3] and rational point 2/3 are explicit. Does 2/3 belong to this closed interval?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"Exact cross multiplication confirms -2≤2/3≤3; equality with an endpoint is included in a closed interval."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0012","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d9dc95ca9627a3f9c16c81e3c99d2a15420a3b4d7619928d69d8c238441ada3","family_id":"qf-real-rational-closed-interval","seed":20261135,"semantic_template":"qf-real-rational-closed-interval/{\"a\":-1,\"b\":5,\"operation\":\"closed_interval\",\"point\":[3,4]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [-1, 5] and rational point 3/4 are explicit. Does 3/4 belong to this closed interval?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"Exact cross multiplication confirms -1≤3/4≤5; equality with an endpoint is included in a closed interval."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0013","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0afb81794913e0bb2c44bf9793baf094f8dff6d476e635906f80817404e3772","family_id":"qf-real-rational-closed-interval","seed":20261136,"semantic_template":"qf-real-rational-closed-interval/{\"a\":0,\"b\":1,\"operation\":\"closed_interval\",\"point\":[0,1]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [0, 1] and rational point 0/1 are explicit. Does 0/1 belong to this closed interval?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"Exact cross multiplication confirms 0≤0/1≤1; equality with an endpoint is included in a closed interval."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0014","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9eef4d0fe2aea132c5fec62a771f06144df559643c895e1653a546fd7bf23b0","family_id":"qf-real-rational-closed-interval","seed":20261137,"semantic_template":"qf-real-rational-closed-interval/{\"a\":1,\"b\":3,\"operation\":\"closed_interval\",\"point\":[1,1]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [1, 3] and rational point 1/1 are explicit. Does 1/1 belong to this closed interval?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"Exact cross multiplication confirms 1≤1/1≤3; equality with an endpoint is included in a closed interval."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0015","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5556e38e449adfd96c8eb3bdc4cb797ec68b5fed14302f9c8b7787f22ef1978","family_id":"qf-real-rational-closed-interval","seed":20261138,"semantic_template":"qf-real-rational-closed-interval/{\"a\":2,\"b\":5,\"operation\":\"closed_interval\",\"point\":[4,3]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [2, 5] and rational point 4/3 are explicit. Does 4/3 belong to this closed interval?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"Exact cross multiplication shows that 4/3 lies outside [2, 5]."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0016","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c2da379813373f44c4a865ae3acd7aef75aec8ad4ec0f8c2f55260314264585","family_id":"qf-real-rational-closed-interval","seed":20261139,"semantic_template":"qf-real-rational-closed-interval/{\"a\":3,\"b\":7,\"operation\":\"closed_interval\",\"point\":[3,4]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [3, 7] and rational point 3/4 are explicit. Does 3/4 belong to this closed interval?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"Exact cross multiplication shows that 3/4 lies outside [3, 7]."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0017","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ebfed7e5eb1f86787a087428d2cd397aa60f1218bb8e04a1248e8f5904d97d3","family_id":"qf-real-rational-closed-interval","seed":20261140,"semantic_template":"qf-real-rational-closed-interval/{\"a\":4,\"b\":9,\"operation\":\"closed_interval\",\"point\":[8,1]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [4, 9] and rational point 8/1 are explicit. Does 8/1 belong to this closed interval?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"Exact cross multiplication confirms 4≤8/1≤9; equality with an endpoint is included in a closed interval."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0018","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9e245b78ae8cac7b1f7916ae876efe971a5275204db701ba541ed2242b82b80","family_id":"qf-real-rational-closed-interval","seed":20261141,"semantic_template":"qf-real-rational-closed-interval/{\"a\":5,\"b\":11,\"operation\":\"closed_interval\",\"point\":[4,1]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [5, 11] and rational point 4/1 are explicit. Does 4/1 belong to this closed interval?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"Exact cross multiplication shows that 4/1 lies outside [5, 11]."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0019","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f8d569fee6433d6d3f47a898424aefd8c2dab98ae93ac77bcc50f9e03982ee0","family_id":"qf-real-rational-closed-interval","seed":20261142,"semantic_template":"qf-real-rational-closed-interval/{\"a\":6,\"b\":7,\"operation\":\"closed_interval\",\"point\":[2,1]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [6, 7] and rational point 2/1 are explicit. Does 2/1 belong to this closed interval?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"Exact cross multiplication shows that 2/1 lies outside [6, 7]."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0020","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2343901013fb428d75260f2387dbc070ba4c77326811255185b91e534834f08","family_id":"qf-real-rational-closed-interval","seed":20261143,"semantic_template":"qf-real-rational-closed-interval/{\"a\":7,\"b\":9,\"operation\":\"closed_interval\",\"point\":[2,1]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [7, 9] and rational point 2/1 are explicit. Does 2/1 belong to this closed interval?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"Exact cross multiplication shows that 2/1 lies outside [7, 9]."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0021","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0670742023c43cb2ca2bdc54d4586cf9e4c01d90d1578b7b900372c4e205e9e","family_id":"qf-real-rational-closed-interval","seed":20261144,"semantic_template":"qf-real-rational-closed-interval/{\"a\":8,\"b\":11,\"operation\":\"closed_interval\",\"point\":[8,1]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [8, 11] and rational point 8/1 are explicit. Does 8/1 belong to this closed interval?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"Exact cross multiplication confirms 8≤8/1≤11; equality with an endpoint is included in a closed interval."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0022","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7042455d1c73af3e596d621a303fb58dd62f7895fbb01c62926e1eb1f053863","family_id":"qf-real-rational-closed-interval","seed":20261145,"semantic_template":"qf-real-rational-closed-interval/{\"a\":9,\"b\":13,\"operation\":\"closed_interval\",\"point\":[5,1]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [9, 13] and rational point 5/1 are explicit. Does 5/1 belong to this closed interval?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"Exact cross multiplication shows that 5/1 lies outside [9, 13]."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0023","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1be94bfa6b1f06ecb08c29ac8dabc2d14ca344b904952baaeab00146443f2bdd","family_id":"qf-real-rational-closed-interval","seed":20261146,"semantic_template":"qf-real-rational-closed-interval/{\"a\":10,\"b\":15,\"operation\":\"closed_interval\",\"point\":[14,3]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [10, 15] and rational point 14/3 are explicit. Does 14/3 belong to this closed interval?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"Exact cross multiplication shows that 14/3 lies outside [10, 15]."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0024","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bac0b74720cea3740982aeaeaf64c5401b2ddb95d251696f30f6c6851dcc294","family_id":"qf-real-rational-closed-interval","seed":20261147,"semantic_template":"qf-real-rational-closed-interval/{\"a\":11,\"b\":17,\"operation\":\"closed_interval\",\"point\":[13,4]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [11, 17] and rational point 13/4 are explicit. Does 13/4 belong to this closed interval?","answer":{"kind":"choice","value":"No","choices":["Yes","No"],"explanation":"Exact cross multiplication shows that 13/4 lies outside [11, 17]."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-closed-interval-0025","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dece1e3184d4f14e5e404a12a2dc8dacc8a9574ee3c95035476b1e67084e3e01","family_id":"qf-real-rational-closed-interval","seed":20261148,"semantic_template":"qf-real-rational-closed-interval/{\"a\":12,\"b\":13,\"operation\":\"closed_interval\",\"point\":[12,1]}"},"fields":{"title":"A Rational Point in a Closed Interval","stem":"The closed interval [12, 13] and rational point 12/1 are explicit. Does 12/1 belong to this closed interval?","answer":{"kind":"choice","value":"Yes","choices":["Yes","No"],"explanation":"Exact cross multiplication confirms 12≤12/1≤13; equality with an endpoint is included in a closed interval."},"skills":["closed intervals","rational comparison"],"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-20260923-real-rational-point-distance-0001","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0c91473df03227cbd0be1f3ceb3879787449ef7464a41d23f6438343f831a97","family_id":"qf-real-rational-point-distance","seed":20261224,"semantic_template":"qf-real-rational-point-distance/{\"left\":[-10,1],\"operation\":\"exact_rational_distance\",\"right\":[12,1]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are -10/1 and 12/1. Find the exact distance |-10/1−12/1| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"22/1","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 22/1."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0002","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8e92afd9b554dcbf6754f91c9c5d0acfa03399aee671e6b44dfded92679c488","family_id":"qf-real-rational-point-distance","seed":20261225,"semantic_template":"qf-real-rational-point-distance/{\"left\":[-9,2],\"operation\":\"exact_rational_distance\",\"right\":[11,2]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are -9/2 and 11/2. Find the exact distance |-9/2−11/2| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"10/1","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 10/1."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0003","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5331d6803f8a85cd2172e84da20102fc8c46e6a4c741031c82c0e847b91d251a","family_id":"qf-real-rational-point-distance","seed":20261226,"semantic_template":"qf-real-rational-point-distance/{\"left\":[-8,3],\"operation\":\"exact_rational_distance\",\"right\":[10,3]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are -8/3 and 10/3. Find the exact distance |-8/3−10/3| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"6/1","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 6/1."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0004","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f8124755b2c40360811bba8f24ebad9fa3ca75a90b29f64065bf41b88a1e517","family_id":"qf-real-rational-point-distance","seed":20261227,"semantic_template":"qf-real-rational-point-distance/{\"left\":[-7,4],\"operation\":\"exact_rational_distance\",\"right\":[9,4]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are -7/4 and 9/4. Find the exact distance |-7/4−9/4| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"4/1","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 4/1."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0005","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"251294001b2645bc8cb42d17a371a2b29c352e12e082229a1cebd045ec549d1d","family_id":"qf-real-rational-point-distance","seed":20261228,"semantic_template":"qf-real-rational-point-distance/{\"left\":[-6,5],\"operation\":\"exact_rational_distance\",\"right\":[8,5]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are -6/5 and 8/5. Find the exact distance |-6/5−8/5| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"14/5","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 14/5."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0006","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8eae0cb98f8167d39b903da407246fa6ff49ceade462a00a3f9135c518cb07b0","family_id":"qf-real-rational-point-distance","seed":20261229,"semantic_template":"qf-real-rational-point-distance/{\"left\":[-5,1],\"operation\":\"exact_rational_distance\",\"right\":[7,6]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are -5/1 and 7/6. Find the exact distance |-5/1−7/6| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"37/6","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 37/6."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0007","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad041f6b57e3cb58d9ae147c06681183b953ee96c16a46fb0be18fdf30c3c28a","family_id":"qf-real-rational-point-distance","seed":20261230,"semantic_template":"qf-real-rational-point-distance/{\"left\":[-2,1],\"operation\":\"exact_rational_distance\",\"right\":[6,7]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are -2/1 and 6/7. Find the exact distance |-2/1−6/7| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"20/7","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 20/7."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0008","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d008ee1e3eb5f45adf850300a1217c5c2d1ecb89a4816629572e36f3a251134","family_id":"qf-real-rational-point-distance","seed":20261231,"semantic_template":"qf-real-rational-point-distance/{\"left\":[-1,1],\"operation\":\"exact_rational_distance\",\"right\":[5,1]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are -1/1 and 5/1. Find the exact distance |-1/1−5/1| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"6/1","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 6/1."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0009","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5682f66833fbec5b6be40f3fa2003b93d469ad5bb58e41ddb06fd853a2d69010","family_id":"qf-real-rational-point-distance","seed":20261232,"semantic_template":"qf-real-rational-point-distance/{\"left\":[-1,2],\"operation\":\"exact_rational_distance\",\"right\":[2,1]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are -1/2 and 2/1. Find the exact distance |-1/2−2/1| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"5/2","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 5/2."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0010","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7895eb10771fc83ee40b4f74a070fbb43b3bf2806b8f33e1a6051d9c1f8fa37","family_id":"qf-real-rational-point-distance","seed":20261233,"semantic_template":"qf-real-rational-point-distance/{\"left\":[-1,5],\"operation\":\"exact_rational_distance\",\"right\":[1,1]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are -1/5 and 1/1. Find the exact distance |-1/5−1/1| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"6/5","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 6/5."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0011","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"336ddd52ca96171508cc1ee4dcb63a6442f2356d6e02736afed2a5586c21a563","family_id":"qf-real-rational-point-distance","seed":20261234,"semantic_template":"qf-real-rational-point-distance/{\"left\":[0,1],\"operation\":\"exact_rational_distance\",\"right\":[1,2]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are 0/1 and 1/2. Find the exact distance |0/1−1/2| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"1/2","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 1/2."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0012","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9069fdd4d988e4f1b01d40272312746ed5aca4ac70b4294aa79db9e74546174a","family_id":"qf-real-rational-point-distance","seed":20261235,"semantic_template":"qf-real-rational-point-distance/{\"left\":[1,2],\"operation\":\"exact_rational_distance\",\"right\":[1,5]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are 1/2 and 1/5. Find the exact distance |1/2−1/5| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"3/10","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 3/10."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0013","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41b069591c3cff1bc5b39c829bc149b38dc93aec83e99eabd67d12b24633cca2","family_id":"qf-real-rational-point-distance","seed":20261236,"semantic_template":"qf-real-rational-point-distance/{\"left\":[2,3],\"operation\":\"exact_rational_distance\",\"right\":[0,1]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are 2/3 and 0/1. Find the exact distance |2/3−0/1| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"2/3","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 2/3."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0014","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34509d72d4559fef919a673f92794e4af60a7444d8dc0f9586e85bd2ea7f8d46","family_id":"qf-real-rational-point-distance","seed":20261237,"semantic_template":"qf-real-rational-point-distance/{\"left\":[3,4],\"operation\":\"exact_rational_distance\",\"right\":[-1,7]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are 3/4 and -1/7. Find the exact distance |3/4−-1/7| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"25/28","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 25/28."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0015","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"511257dd5eb422d6a48452ef4fb62a6b7eb397af1ecf8c755e94b2dd1da73cf7","family_id":"qf-real-rational-point-distance","seed":20261238,"semantic_template":"qf-real-rational-point-distance/{\"left\":[4,5],\"operation\":\"exact_rational_distance\",\"right\":[-2,1]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are 4/5 and -2/1. Find the exact distance |4/5−-2/1| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"14/5","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 14/5."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0016","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e275c4a96c03b2b8a338b67cc06393615719923465d324eedd350bb5eb2fb905","family_id":"qf-real-rational-point-distance","seed":20261239,"semantic_template":"qf-real-rational-point-distance/{\"left\":[5,1],\"operation\":\"exact_rational_distance\",\"right\":[-3,2]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are 5/1 and -3/2. Find the exact distance |5/1−-3/2| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"13/2","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 13/2."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0017","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cb539890ea43986ef68b3ecd6300926c191ad66b83eb49d624cfc31a610c992","family_id":"qf-real-rational-point-distance","seed":20261240,"semantic_template":"qf-real-rational-point-distance/{\"left\":[3,1],\"operation\":\"exact_rational_distance\",\"right\":[-4,3]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are 3/1 and -4/3. Find the exact distance |3/1−-4/3| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"13/3","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 13/3."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0018","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"024246c6dedcdbfdce0171911b72fce5ae6b924857a80810934139543ac60089","family_id":"qf-real-rational-point-distance","seed":20261241,"semantic_template":"qf-real-rational-point-distance/{\"left\":[7,3],\"operation\":\"exact_rational_distance\",\"right\":[-5,4]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are 7/3 and -5/4. Find the exact distance |7/3−-5/4| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"43/12","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 43/12."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0019","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f02bfb0b0a9c1813b89cae0081b181580a89024096e9db4ce6922b64de2fdda3","family_id":"qf-real-rational-point-distance","seed":20261242,"semantic_template":"qf-real-rational-point-distance/{\"left\":[2,1],\"operation\":\"exact_rational_distance\",\"right\":[-6,5]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are 2/1 and -6/5. Find the exact distance |2/1−-6/5| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"16/5","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 16/5."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0020","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc7703af4557b0df67525b45f10ae713d77c4424b8f4478b7d4006611c52de7e","family_id":"qf-real-rational-point-distance","seed":20261243,"semantic_template":"qf-real-rational-point-distance/{\"left\":[9,5],\"operation\":\"exact_rational_distance\",\"right\":[-7,6]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are 9/5 and -7/6. Find the exact distance |9/5−-7/6| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"89/30","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 89/30."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0021","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04701ce1634fc5b612ae63a0a37c95aeadd53a34371aad0ee49fce635225e04b","family_id":"qf-real-rational-point-distance","seed":20261244,"semantic_template":"qf-real-rational-point-distance/{\"left\":[10,1],\"operation\":\"exact_rational_distance\",\"right\":[-8,7]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are 10/1 and -8/7. Find the exact distance |10/1−-8/7| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"78/7","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 78/7."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0022","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d83e0595c28c7ee8ed654cd31078d73be9acc5fe94821244513ee775177a8dd8","family_id":"qf-real-rational-point-distance","seed":20261245,"semantic_template":"qf-real-rational-point-distance/{\"left\":[11,2],\"operation\":\"exact_rational_distance\",\"right\":[-9,1]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are 11/2 and -9/1. Find the exact distance |11/2−-9/1| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"29/2","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 29/2."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0023","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2c3f88148611b04af113f7632b8b9233c01044c4d6183113af0fcb17c32999b","family_id":"qf-real-rational-point-distance","seed":20261246,"semantic_template":"qf-real-rational-point-distance/{\"left\":[4,1],\"operation\":\"exact_rational_distance\",\"right\":[-5,1]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are 4/1 and -5/1. Find the exact distance |4/1−-5/1| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"9/1","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 9/1."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0024","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b79e75752b6ec9c9388ec2db6f011454cdd0e0a1cc443450681ba3763e53ed3","family_id":"qf-real-rational-point-distance","seed":20261247,"semantic_template":"qf-real-rational-point-distance/{\"left\":[13,4],\"operation\":\"exact_rational_distance\",\"right\":[-11,3]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are 13/4 and -11/3. Find the exact distance |13/4−-11/3| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"83/12","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 83/12."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20260923-real-rational-point-distance-0025","batch_id":"batch-20260923-real-number-rational-points","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b4ac4b44ae952dd39fe69a804e51a32cbbd15bc66163d30e25368415425760e","family_id":"qf-real-rational-point-distance","seed":20261248,"semantic_template":"qf-real-rational-point-distance/{\"left\":[14,5],\"operation\":\"exact_rational_distance\",\"right\":[-3,1]}"},"fields":{"title":"Exact Distance Between Rational Points","stem":"Two rational real points are 14/5 and -3/1. Find the exact distance |14/5−-3/1| as a lowest-terms fraction.","answer":{"kind":"short_text","value":"29/5","explanation":"Subtract the integer numerator-denominator pairs exactly and take the absolute value, giving 29/5."},"skills":["rational numbers","absolute difference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}