{"id":"cb_question_bank:b510c444-9f16-4a2b-8b9c-a98cb0c0b332","source_id":"cb_question_bank","source_item_id":"b510c444-9f16-4a2b-8b9c-a98cb0c0b332","external_id":"7c3cceff-5961-4825-a23b-44ce88420238","ibn":null,"question_id":"20291f47","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.A.","skill":"Equivalent expressions","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Which expression is equivalent to StartFraction y plus 12 Over x minus 8 EndFraction plus StartFraction y left parenthesis x minus 8 right parenthesis Over x squared y minus 8 x y EndFraction ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">Which expression is equivalent to&nbsp;<math alttext=\"StartFraction y plus 12 Over x minus 8 EndFraction plus StartFraction y left parenthesis x minus 8 right parenthesis Over x squared y minus 8 x y EndFraction\"><mfrac><mrow><mi>y</mi><mo>+</mo><mrow><mn>12</mn></mrow></mrow><mrow><mi>x</mi><mo>-</mo><mrow><mn>8</mn></mrow></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>y</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mrow><mn>8</mn></mrow></mrow></mfenced></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><mo>-</mo><mrow><mn>8</mn></mrow><mi>x</mi><mi>y</mi></mrow></mfrac></math>?</p>","rationale_html":"<p>Choice C is correct. Factoring the denominator in the second term of the given expression gives <math alttext=\"StartFraction y plus 12 Over x minus 8 EndFraction plus StartFraction y left parenthesis x minus 8 right parenthesis Over x y left parenthesis x minus 8 right parenthesis EndFraction\"><mfrac><mrow><mi>y</mi><mo>+</mo><mn>12</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>y</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfenced></mrow><mrow><mi>x</mi><mi>y</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfenced></mrow></mfrac></math>. This expression can be rewritten with common denominators by multiplying the first term by <math alttext=\"StartFraction x y Over x y EndFraction\"><mfrac><mrow><mi>x</mi><mi>y</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></mfrac></math>, giving&nbsp;<math alttext=\"StartFraction x y left parenthesis y plus 12 right parenthesis Over x y left parenthesis x minus 8 right parenthesis EndFraction plus StartFraction y left parenthesis x minus 8 right parenthesis Over x y left parenthesis x minus 8 right parenthesis EndFraction\"><mfrac><mrow><mi>x</mi><mi>y</mi><mfenced><mrow><mi>y</mi><mo>+</mo><mn>12</mn></mrow></mfenced></mrow><mrow><mi>x</mi><mi>y</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>y</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfenced></mrow><mrow><mi>x</mi><mi>y</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfenced></mrow></mfrac></math>. Adding these two terms yields <math alttext=\"StartFraction x y left parenthesis y plus 12 right parenthesis plus y left parenthesis x minus 8 right parenthesis Over x y left parenthesis x minus 8 right parenthesis EndFraction\"><mfrac><mrow><mi>x</mi><mi>y</mi><mfenced><mrow><mi>y</mi><mo>+</mo><mn>12</mn></mrow></mfenced><mo>+</mo><mi>y</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfenced></mrow><mrow><mi>x</mi><mi>y</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfenced></mrow></mfrac></math>. Using the distributive property to rewrite this expression gives <math alttext=\"StartFraction x y squared plus 12 x y plus x y minus 8 y Over x squared y minus 8 x y EndFraction\"><mfrac><mrow><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mi>y</mi><mo>+</mo><mi>x</mi><mi>y</mi><mo>-</mo><mn>8</mn><mi>y</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><mo>-</mo><mn>8</mn><mi>x</mi><mi>y</mi></mrow></mfrac></math>. Combining the like terms in the numerator of this expression gives <math alttext=\"StartFraction x y squared plus 13 x y minus 8 y Over x squared y minus 8 x y EndFraction\"><mfrac><mrow><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>13</mn><mi>x</mi><mi>y</mi><mo>-</mo><mn>8</mn><mi>y</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><mo>-</mo><mn>8</mn><mi>x</mi><mi>y</mi></mrow></mfrac></math>.</p>\n<p>Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["C"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.821000+00:00","source_updated_at":"2023-08-02T20:25:59.821000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"StartFraction x y plus y plus 4 Over x cubed y minus 16 x squared y plus 64 x y EndFraction\"><mrow>\n\t<mfrac>\n\t\t<mrow>\n\t\t\t<mrow>\n\t\t\t\t<mi>x</mi>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t\t<mo>+</mo>\n\t\t\t<mi>y</mi>\n\t\t\t<mo>+</mo>\n\t\t\t<mn>4</mn>\n\t\t</mrow>\n\t\t<mrow>\n\t\t\t<mrow>\n\t\t\t\t<msup>\n\t\t\t\t\t<mi>x</mi>\n\t\t\t\t\t<mn>3</mn>\n\t\t\t\t</msup>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t\t<mo>-</mo>\n\t\t\t<mrow>\n\t\t\t\t<mn>16</mn>\n\t\t\t\t<msup>\n\t\t\t\t\t<mi>x</mi>\n\t\t\t\t\t<mn>2</mn>\n\t\t\t\t</msup>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t\t<mo>+</mo>\n\t\t\t<mrow>\n\t\t\t\t<mn>64</mn>\n\t\t\t\t<mi>x</mi>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t</mrow>\n\t</mfrac>\n</mrow>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"StartFraction x y plus 9 y plus 12 Over x squared y minus 8 x y plus x minus 8 EndFraction\"><mrow>\n\t<mfrac>\n\t\t<mrow>\n\t\t\t<mrow>\n\t\t\t\t<mi>x</mi>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t\t<mo>+</mo>\n\t\t\t<mrow>\n\t\t\t\t<mn>9</mn>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t\t<mo>+</mo>\n\t\t\t<mn>12</mn>\n\t\t</mrow>\n\t\t<mrow>\n\t\t\t<mrow>\n\t\t\t\t<msup>\n\t\t\t\t\t<mi>x</mi>\n\t\t\t\t\t<mn>2</mn>\n\t\t\t\t</msup>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t\t<mo>-</mo>\n\t\t\t<mrow>\n\t\t\t\t<mn>8</mn>\n\t\t\t\t<mi>x</mi>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t\t<mo>+</mo>\n\t\t\t<mi>x</mi>\n\t\t\t<mo>-</mo>\n\t\t\t<mn>8</mn>\n\t\t</mrow>\n\t</mfrac>\n</mrow>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"StartFraction x y squared plus 13 x y minus 8 y Over x squared y minus 8 x y EndFraction\"><mrow>\n\t<mfrac>\n\t\t<mrow>\n\t\t\t<mrow>\n\t\t\t\t<mi>x</mi>\n\t\t\t\t<msup>\n\t\t\t\t\t<mi>y</mi>\n\t\t\t\t\t<mn>2</mn>\n\t\t\t\t</msup>\n\t\t\t</mrow>\n\t\t\t<mo>+</mo>\n\t\t\t<mrow>\n\t\t\t\t<mn>13</mn>\n\t\t\t\t<mi>x</mi>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t\t<mo>-</mo>\n\t\t\t<mrow>\n\t\t\t\t<mn>8</mn>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t</mrow>\n\t\t<mrow>\n\t\t\t<mrow>\n\t\t\t\t<msup>\n\t\t\t\t\t<mi>x</mi>\n\t\t\t\t\t<mn>2</mn>\n\t\t\t\t</msup>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t\t<mo>-</mo>\n\t\t\t<mrow>\n\t\t\t\t<mn>8</mn>\n\t\t\t\t<mi>x</mi>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t</mrow>\n\t</mfrac>\n</mrow>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"StartFraction x y squared plus 13 x y minus 8 y Over x cubed y minus 16 x squared y plus 64 x y EndFraction\"><mrow>\n\t<mfrac>\n\t\t<mrow>\n\t\t\t<mrow>\n\t\t\t\t<mi>x</mi>\n\t\t\t\t<msup>\n\t\t\t\t\t<mi>y</mi>\n\t\t\t\t\t<mn>2</mn>\n\t\t\t\t</msup>\n\t\t\t</mrow>\n\t\t\t<mo>+</mo>\n\t\t\t<mrow>\n\t\t\t\t<mn>13</mn>\n\t\t\t\t<mi>x</mi>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t\t<mo>-</mo>\n\t\t\t<mrow>\n\t\t\t\t<mn>8</mn>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t</mrow>\n\t\t<mrow>\n\t\t\t<mrow>\n\t\t\t\t<msup>\n\t\t\t\t\t<mi>x</mi>\n\t\t\t\t\t<mn>3</mn>\n\t\t\t\t</msup>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t\t<mo>-</mo>\n\t\t\t<mrow>\n\t\t\t\t<mn>16</mn>\n\t\t\t\t<msup>\n\t\t\t\t\t<mi>x</mi>\n\t\t\t\t\t<mn>2</mn>\n\t\t\t\t</msup>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t\t<mo>+</mo>\n\t\t\t<mrow>\n\t\t\t\t<mn>64</mn>\n\t\t\t\t<mi>x</mi>\n\t\t\t\t<mi>y</mi>\n\t\t\t</mrow>\n\t\t</mrow>\n\t</mfrac>\n</mrow>\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}}