{"id":"aqua_rat:7f34cbbb360f318ef2b645d0763b04b9","source_id":"aqua_rat","source_item_id":"7f34cbbb360f318ef2b645d0763b04b9","external_id":null,"program":null,"section":"math","category_code":"PHM","category":"Preparing for Higher Math","subcategory_code":null,"subcategory":null,"form_id":null,"passage_id":null,"position":null,"scored":true,"layout_warning":null,"stem":"Suppose for any real number x, [x] denotes the greatest integer less than or equal to x. Let L(x,y) = [x] + [y] + [x + y] and R(x,y) = [2x] + [2y]. Then it is impossible to find any two positive real numbers x and y for which","stimulus":"","stimulus_html":null,"stem_html":"<p>Suppose for any real number x, [x] denotes the greatest integer less than or equal to x. Let L(x,y) = [x] + [y] + [x + y] and R(x,y) = [2x] + [2y]. Then it is impossible to find any two positive real numbers x and y for which</p>","rationale_html":"<p>x – 1≤ [x] ≤ x</p><p>2x + 2y - 3 ≤ L(x,y) ≤ 2x + 2y</p><p>=&gt; a – 3 ≤ L ≤ a</p><p>2x + 2y -2 ≤ L(x,y) ≤ 2x + 2y</p><p>=&gt; a – 2 ≤ R ≤ a</p><p>Therefore, L ≤ R.</p><p>D</p>","correct_answer":["D"],"equiv_key":"b1a62b60a2d22aaaeb63bbbe","has_media":false,"has_mathml":false,"has_table":false,"section_label":"Mathematics","options":[{"label":"A","content_html":"<p>L(x,y) = R(x,y)</p>","ord":0},{"label":"B","content_html":"<p>L(x,y) ≠ R(x,y)</p>","ord":1},{"label":"C","content_html":"<p>L(x,y) &lt; R(x,y)</p>","ord":2},{"label":"D","content_html":"<p>L(x,y) &lt; R(x,y)</p>","ord":3},{"label":"E","content_html":"<p>None</p>","ord":4}],"passage":null,"attribution":{"source_id":"aqua_rat","source_name":"AQuA-RAT algebra word problems with rationales","rights_holder":"Google DeepMind and the AQuA-RAT contributors","canonical_url":"https://github.com/google-deepmind/AQuA","retrieved_from":"https://raw.githubusercontent.com/google-deepmind/AQuA/master/","attribution":"Algebra word problems from the AQuA-RAT dataset (Ling et al., 2017), published by Google DeepMind under the Apache License 2.0.","disclaimer":"AQuA-RAT items are crowdsourced algebra problems, not ACT questions. They carry five answer choices in the pre-2025 ACT Mathematics style and no ACT reporting category. Useful for drilling Preparing for Higher Math content; not a substitute for a real form.","license":"Apache License 2.0. Crowdsourced by DeepMind and published for free use, including commercially.","redistributable":true,"item_count":508},"equivalents":[]}