{"id":"cb_question_bank:48486e3c-cc14-4f67-abfe-313edea866e1","source_id":"cb_question_bank","source_item_id":"48486e3c-cc14-4f67-abfe-313edea866e1","external_id":"7e2ec609-ff10-4d96-85a3-7a8db042e0c0","ibn":null,"question_id":"6c71f3ec","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.E.","skill":"Linear inequalities in one or two variables","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"A salesperson’s total earnings consist of a base salary of x dollars per year, plus commission earnings of 11 percent sign of the total sales the salesperson makes during the year. This year, the salesperson has a goal for the total earnings to be at least 3 times and at most 4 times the base salary. Which of the following inequalities represents all possible values of total sales s , in dollars, the salesperson can make this year in order to meet that goal?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">A salesperson’s total earnings consist of a base salary of <math alttext=\"x\"><mi>x</mi>\n</math> dollars per year, plus commission earnings of <math alttext=\"11 percent sign\"><mn>11</mn>\n<mo>%</mo></math> of the total sales the salesperson makes during the year. This year, the salesperson has a goal for the total earnings to be at least <math alttext=\"3\"><mn>3</mn>\n</math> times and at most <math alttext=\"4\"><mn>4</mn>\n</math> times the base salary. Which of the following inequalities represents all possible values of total sales <math alttext=\"s\"><mi>s</mi>\n</math>, in dollars, the salesperson can make this year in order to meet that goal?&nbsp;</p>","rationale_html":"<p style=\"text-align: left;\">Choice B is correct. It’s given that a salesperson's total earnings consist of a base salary of <math alttext=\"x\"><mi>x</mi>\n</math> dollars per year plus commission earnings of <math alttext=\"11 percent sign\"><mn>11</mn><mo>%</mo></math> of the total sales the salesperson makes during the year. If the salesperson makes <math alttext=\"s\"><mi>s</mi>\n</math> dollars in total sales this year, the salesperson’s total earnings can be represented by the expression <math alttext=\"x plus 0.11 s\"><mi>x</mi><mo>+</mo><mn>0.11</mn><mi>s</mi></math>. It’s also given that the salesperson has a goal for the total earnings to be at least <math alttext=\"3\"><mn>3</mn>\n</math> times and at most <math alttext=\"4\"><mn>4</mn>\n</math> times the base salary, which can be represented by the expressions <math alttext=\"3 x\"><mrow>\n\t<mn>3</mn>\n\t<mi>x</mi>\n</mrow>\n</math> and <math alttext=\"4 x\"><mrow>\n\t<mn>4</mn>\n\t<mi>x</mi>\n</mrow>\n</math>, respectively. Therefore, this situation can be represented by the inequality <math alttext=\"3 x less than or equals x plus 0.11 s less than or equals 4 x\"><mn>3</mn><mi>x</mi><mo>≤</mo><mi>x</mi><mo>+</mo><mn>0.11</mn><mi>s</mi><mo>≤</mo><mn>4</mn><mi>x</mi></math>. Subtracting <math alttext=\"x\"><mi>x</mi>\n</math> from each part of this inequality yields <math alttext=\"2 x less than or equals 0.11 s less than or equals 3 x\"><mn>2</mn><mi>x</mi><mo>≤</mo><mn>0.11</mn><mi>s</mi><mo>≤</mo><mn>3</mn><mi>x</mi></math>. Dividing each part of this inequality by <math alttext=\"0.11\"><mn>0.11</mn>\n</math> yields <math alttext=\"StartFraction 2 Over 0.11 EndFraction x less than or equals s less than or equals StartFraction 3 Over 0.11 EndFraction x\"><mfrac><mn>2</mn><mn>0.11</mn></mfrac><mi>x</mi><mo>≤</mo><mi>s</mi><mo>≤</mo><mfrac><mn>3</mn><mn>0.11</mn></mfrac><mi>x</mi></math>. Therefore, the inequality <math alttext=\"StartFraction 2 Over 0.11 EndFraction x less than or equals s less than or equals StartFraction 3 Over 0.11 EndFraction x\"><mfrac><mn>2</mn><mn>0.11</mn></mfrac><mi>x</mi><mo>≤</mo><mi>s</mi><mo>≤</mo><mfrac><mn>3</mn><mn>0.11</mn></mfrac><mi>x</mi></math> represents all possible values of total sales <math alttext=\"s\"><mi>s</mi>\n</math>, in dollars, the salesperson can make this year in order to meet their goal.</p>\n<p style=\"text-align: left;\">Choice A is incorrect. This inequality represents a situation in which the total sales, rather than the total earnings, are at least <math alttext=\"2\"><mn>2</mn>\n</math> times and at most <math alttext=\"3\"><mn>3</mn>\n</math> times, rather than at least <math alttext=\"3\"><mn>3</mn>\n</math> times and at most <math alttext=\"4\"><mn>4</mn>\n</math> times, the base salary.</p>\n<p style=\"text-align: left;\">Choice C is incorrect. This inequality represents a situation in which the total sales, rather than the total earnings, are at least <math alttext=\"3\"><mn>3</mn>\n</math> times and at most <math alttext=\"4\"><mn>4</mn>\n</math> times the base salary.</p>\n<p style=\"text-align: left;\">Choice D is incorrect. This inequality represents a situation in which the total earnings are at least <math alttext=\"4\"><mn>4</mn>\n</math> times and at most <math alttext=\"5\"><mn>5</mn>\n</math> times, rather than at least <math alttext=\"3\"><mn>3</mn>\n</math> times and at most <math alttext=\"4\"><mn>4</mn>\n</math> times, the base salary.</p>","correct_answer":["B"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.820000+00:00","source_updated_at":"2023-08-02T20:25:59.820000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"2 x less than or equals s less than or equals 3 x\"><mrow>\n\t<mn>2</mn>\n\t<mi>x</mi>\n</mrow>\n<mo>≤</mo><mi>s</mi><mo>≤</mo><mrow>\n\t<mn>3</mn>\n\t<mi>x</mi>\n</mrow>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"StartFraction 2 Over 0.11 EndFraction x less than or equals s less than or equals StartFraction 3 Over 0.11 EndFraction x\"><mfrac><mrow><mn>2</mn></mrow><mrow><mn>0.11</mn></mrow></mfrac><mi>x</mi><mo>≤</mo><mi>s</mi><mo>≤</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>0.11</mn></mrow></mfrac><mi>x</mi></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"3 x less than or equals s less than or equals 4 x\"><mrow>\n\t<mn>3</mn>\n\t<mi>x</mi>\n</mrow>\n<mo>≤</mo><mi>s</mi><mo>≤</mo><mrow>\n\t<mn>4</mn>\n\t<mi>x</mi>\n</mrow>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"StartFraction 3 Over 0.11 EndFraction x less than or equals s less than or equals StartFraction 4 Over 0.11 EndFraction x\"><mfrac><mrow><mn>3</mn></mrow><mrow><mn>0.11</mn></mrow></mfrac><mi>x</mi><mo>≤</mo><mi>s</mi><mo>≤</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>0.11</mn></mrow></mfrac><mi>x</mi></math></p>","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}}