{"id":"cb_question_bank:016d3534-2566-4551-af72-a61ad0c95b5f","source_id":"cb_question_bank","source_item_id":"016d3534-2566-4551-af72-a61ad0c95b5f","external_id":null,"ibn":"022222-DC","question_id":"f224df07","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.E.","skill":"Linear inequalities in one or two variables","difficulty":"M","score_band":4,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"A cargo helicopter delivers only 100-pound packages and 120-pound packages. For each delivery trip, the helicopter must carry at least 10 packages, and the total weight of the packages can be at most 1,100 pounds. What is the maximum number of 120-pound packages that the helicopter can carry per trip?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">A cargo helicopter delivers only 100-pound packages and 120-pound packages. For each delivery trip, the helicopter must carry at least 10 packages, and the total weight of the packages can be at most 1,100 pounds. What is the maximum number of 120-pound packages that the helicopter can carry per trip?</p>\n","rationale_html":"<p>Choice C is correct. Let <span class=\"italic\">a</span> equal the number of 120-pound packages, and let <span class=\"italic\">b</span> equal the number of 100-pound packages. It’s given that the total weight of the packages can be at most 1,100 pounds: the inequality <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6fa7260c2b89a7dc21e1c984262ad8e03697073db5a59fb35aadc17515a8d300\" alt=\"120 a, plus 100 b, is less than or equal to 1,100\"></span> represents this situation. It’s also given that the helicopter must carry at least 10 packages: the inequality <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/58f75349fa607a6d4980dbec697f289925ee211020431514d4b5452754439148\" alt=\"a, plus b, is greater than or equal to 10\"></span> represents this situation. Values of <span class=\"italic\">a</span> and <span class=\"italic\">b</span> that satisfy these two inequalities represent the allowable numbers of 120-pound packages and 100-pound packages the helicopter can transport. To maximize the number of 120-pound packages, <span class=\"italic\">a</span>, in the helicopter, the number of 100-pound packages, <span class=\"italic\">b</span>, in the helicopter needs to be minimized. Expressing <span class=\"italic\">b</span> in terms of <span class=\"italic\">a</span> in the second inequality yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e322a24a503f14dda1a2469ab37e0f6f59d194fa1a2bdfdc2d4b0ebbe8d5af72\" alt=\"b is greater than or equal to, 10 minus a\"></span>, so the minimum value of <span class=\"italic\">b</span> is equal to <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/a88dae479375bfa8c87b8da27c8339b64e21051a99c477d302ba3b7b2f3125f3\" alt=\"10 minus a\"></span>. Substituting <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/a88dae479375bfa8c87b8da27c8339b64e21051a99c477d302ba3b7b2f3125f3\" alt=\"10 minus a\"></span> for <span class=\"italic\">b</span> in the first inequality results in <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1c960ea3eeaa28359df664d881af0590ccb9039e36aaf840c57f61141c1bb0d0\" alt=\"120 a, plus 100, times, open parenthesis, 10 minus a, close parenthesis, is less than or equal to 1,100\"></span>. Using the distributive property to rewrite this inequality yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c208bdee5338a394f6e24b4ab5da45761f2fe3cda1840ea8956aa31f5553cc63\" alt=\"120 a, plus 1,000, minus 100 a, is less than or equal to 1,100\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ec99a2493ac61a78744d73a80ba1c5111466a19e94fe382dff67221e63a07825\" alt=\"20 a, plus 1,000, is less than or equal to 1,100\"></span>. Subtracting 1,000 from both sides of this inequality yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6606258baa66fd6a49434321d3fac2fcfd85eb9554bdf774157c5f43cc4bae7a\" alt=\"20 a, is less than or equal to 100\"></span>. Dividing both sides of this inequality by 20 results in <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/170f12c1dad093042d60ec22dcf9ec74900b65241ccc1a2a40ffdb0aa0735827\" alt=\"a, is less than or equal to 5\"></span>. This means that the maximum number of 120-pound packages that the helicopter can carry per trip is 5.</p><p>Choices A, B, and D are incorrect and may result from incorrectly creating or solving the system of inequalities.</p><p></p>\n","correct_answer":["C"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.617000+00:00","source_updated_at":"2023-08-02T20:25:59.617000+00:00","options":[{"label":"A","content_html":"<p class=\"choice_paragraph \">2</p>\n","ord":0},{"label":"B","content_html":"<p class=\"choice_paragraph \">4</p>\n","ord":1},{"label":"C","content_html":"<p class=\"choice_paragraph \">5</p>\n","ord":2},{"label":"D","content_html":"<p class=\"choice_paragraph \">6</p>\n","ord":3}],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}