{"id":"cb_question_bank:5bd1a41a-9b67-4dd6-b71a-537ed462b6f8","source_id":"cb_question_bank","source_item_id":"5bd1a41a-9b67-4dd6-b71a-537ed462b6f8","external_id":"a8304c28-11ca-4f4b-b7bf-4379eab3e67e","ibn":null,"question_id":"203774bc","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"M","score_band":5,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The product of two positive integers is 546 . If the first integer is 11 greater than twice the second integer, what is the smaller of the two integers?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">The product of two positive integers is <math alttext=\"546\"><mn>546</mn>\n</math>. If the first integer is <math alttext=\"11\"><mn>11</mn>\n</math> greater than twice the second integer, what is the smaller of the two integers?</p>","rationale_html":"<p style=\"text-align: left;\">Choice B is correct. Let <math alttext=\"x\"><mi>x</mi>\n</math> be the first integer and let <math alttext=\"y\"><mi>y</mi>\n</math> be the second integer. If the first integer is <math alttext=\"11\"><mn>11</mn>\n</math> greater than twice the second integer, then <math alttext=\"x equals 2 y plus 11\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>2</mn>\n\t\t\t<mi>y</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>11</mn>\n\t</mrow>\n</mrow>\n</math>. If the product of the two integers is <math alttext=\"546\"><mn>546</mn>\n</math>, then <math alttext=\"x y equals 546\"><mi>x</mi><mi>y</mi><mo>=</mo><mn>546</mn></math>. Substituting <math alttext=\"2 y plus 11\"><mrow>\n\t<mrow>\n\t\t<mn>2</mn>\n\t\t<mi>y</mi>\n\t</mrow>\n\t<mo>+</mo>\n\t<mn>11</mn>\n</mrow>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> in this equation results in <math alttext=\"left parenthesis 2 y plus 11 right parenthesis y equals 546\"><mfenced><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mn>11</mn></mrow></mfenced><mi>y</mi><mo>=</mo><mn>546</mn></math>. Distributing the <math alttext=\"y\"><mi>y</mi>\n</math> to both terms in the parentheses results in <math alttext=\"2 y squared plus 11 y equals 546\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>2</mn>\n\t\t\t<msup>\n\t\t\t\t<mi>y</mi>\n\t\t\t\t<mn>2</mn>\n\t\t\t</msup>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mrow>\n\t\t\t<mn>11</mn>\n\t\t\t<mi>y</mi>\n\t\t</mrow>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>546</mn>\n</mrow>\n</math>. Subtracting <math alttext=\"546\"><mn>546</mn>\n</math> from both sides of this equation results in <math alttext=\"2 y squared plus 11 y minus 546 equals 0\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>2</mn>\n\t\t\t<msup>\n\t\t\t\t<mi>y</mi>\n\t\t\t\t<mn>2</mn>\n\t\t\t</msup>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mrow>\n\t\t\t<mn>11</mn>\n\t\t\t<mi>y</mi>\n\t\t</mrow>\n\t\t<mo>-</mo>\n\t\t<mn>546</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math>. The left-hand side of this equation can be factored by finding two values whose product is <math alttext=\"2 left parenthesis negative 546 right parenthesis\"><mn>2</mn><mfenced><mrow><mo>-</mo><mn>546</mn></mrow></mfenced></math>, or <math alttext=\"negative 1,092\"><mo>-</mo><mn>1,092</mn>\n</math>, and whose sum is <math alttext=\"11\"><mn>11</mn>\n</math>. The two values whose product is <math alttext=\"negative 1,092\"><mo>-</mo><mn>1,092</mn>\n</math> and whose sum is <math alttext=\"11\"><mn>11</mn>\n</math> are <math alttext=\"39\"><mn>39</mn>\n</math> and <math alttext=\"negative 28\"><mo>-</mo><mn>28</mn>\n</math>. Thus, the equation <math alttext=\"2 y squared plus 11 y minus 546 equals 0\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>2</mn>\n\t\t\t<msup>\n\t\t\t\t<mi>y</mi>\n\t\t\t\t<mn>2</mn>\n\t\t\t</msup>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mrow>\n\t\t\t<mn>11</mn>\n\t\t\t<mi>y</mi>\n\t\t</mrow>\n\t\t<mo>-</mo>\n\t\t<mn>546</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math> can be rewritten as <math alttext=\"2 y squared plus 28 y minus 39 y minus 546 equals 0\"><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>28</mn><mi>y</mi><mo>-</mo><mn>39</mn><mi>y</mi><mo>-</mo><mn>546</mn><mo>=</mo><mn>0</mn></math>, which is equivalent to <math alttext=\"2 y left parenthesis y minus 14 right parenthesis plus 39 left parenthesis y minus 14 right parenthesis equals 0\"><mn>2</mn><mi>y</mi><mfenced><mrow><mi>y</mi><mo>-</mo><mn>14</mn></mrow></mfenced><mo>+</mo><mn>39</mn><mfenced><mrow><mi>y</mi><mo>-</mo><mn>14</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math>, or <math alttext=\"left parenthesis 2 y plus 39 right parenthesis left parenthesis y minus 14 right parenthesis equals 0\"><mfenced><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mn>39</mn></mrow></mfenced><mfenced><mrow><mi>y</mi><mo>-</mo><mn>14</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math>. By the zero product property, it follows that <math alttext=\"2 y plus 39 equals 0\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>2</mn>\n\t\t\t<mi>y</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>39</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math> and <math alttext=\"y minus 14 equals 0\"><mrow>\n\t<mrow>\n\t\t<mi>y</mi>\n\t\t<mo>-</mo>\n\t\t<mn>14</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math>. Subtracting <math alttext=\"39\"><mn>39</mn>\n</math> from both sides of the equation <math alttext=\"2 y plus 39 equals 0\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>2</mn>\n\t\t\t<mi>y</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>39</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math> yields <math alttext=\"2 y equals negative 39\"><mrow>\n\t<mrow>\n\t\t<mn>2</mn>\n\t\t<mi>y</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>39</mn>\n</mrow>\n</math>. Dividing both sides of this equation by <math alttext=\"2\"><mn>2</mn>\n</math> yields <math alttext=\"y equals negative StartFraction 39 Over 2 EndFraction\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mo>-</mo>\n\t\t<mfrac>\n\t\t\t<mn>39</mn>\n\t\t\t<mn>2</mn>\n\t\t</mfrac>\n\t</mrow>\n</mrow>\n</math>. Since <math alttext=\"y\"><mi>y</mi>\n</math> is a positive integer, the value of <math alttext=\"y\"><mi>y</mi>\n</math> is not <math alttext=\"negative StartFraction 39 Over 2 EndFraction\"><mrow>\n\t<mo>-</mo>\n\t<mfrac>\n\t\t<mn>39</mn>\n\t\t<mn>2</mn>\n\t</mfrac>\n</mrow>\n</math>. Adding <math alttext=\"14\"><mn>14</mn>\n</math> to both sides of the equation <math alttext=\"y minus 14 equals 0\"><mrow>\n\t<mrow>\n\t\t<mi>y</mi>\n\t\t<mo>-</mo>\n\t\t<mn>14</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math> yields <math alttext=\"y equals 14\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mn>14</mn>\n</mrow>\n</math>. Substituting <math alttext=\"14\"><mn>14</mn>\n</math> for <math alttext=\"y\"><mi>y</mi>\n</math> in the equation <math alttext=\"x y equals 546\"><mi>x</mi><mi>y</mi><mo>=</mo><mn>546</mn></math> yields <math alttext=\"x left parenthesis 14 right parenthesis equals 546\"><mi>x</mi><mfenced><mn>14</mn></mfenced><mo>=</mo><mn>546</mn></math>. Dividing both sides of this equation by <math alttext=\"14\"><mn>14</mn>\n</math> results in <math alttext=\"x equals 39\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>39</mn>\n</mrow>\n</math>. Therefore, the two integers are <math alttext=\"14\"><mn>14</mn>\n</math> and <math alttext=\"39\"><mn>39</mn>\n</math>, so the smaller of the two integers is <math alttext=\"14\"><mn>14</mn>\n</math>.</p>\n<p style=\"text-align: left;\">Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice C is incorrect. This is the larger of the two integers.</p>\n<p style=\"text-align: left;\">Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["B"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.825000+00:00","source_updated_at":"2023-08-02T20:25:59.825000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"7\"><mn>7</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"14\"><mn>14</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"39\"><mn>39</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"78\"><mn>78</mn>\n</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}}