{"id":"cb_question_bank:2350e4d0-5363-4994-aa2d-ef1dad6202d6","source_id":"cb_question_bank","source_item_id":"2350e4d0-5363-4994-aa2d-ef1dad6202d6","external_id":"74a8d55a-4db1-41ce-9d72-b5498e6a1d89","ibn":null,"question_id":"b4a6ed81","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.A.","skill":"Equivalent expressions","difficulty":"M","score_band":4,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The expression 90 y Superscript 5 minus 54 y Superscript 4 is equivalent to r y Superscript 4 Baseline left parenthesis 15 y minus 9 right parenthesis , where r is a constant. What is the value of r ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">The expression <math alttext=\"90 y Superscript 5 minus 54 y Superscript 4\"><mrow>\n\t<mrow>\n\t\t<mn>90</mn>\n\t\t<msup>\n\t\t\t<mi>y</mi>\n\t\t\t<mn>5</mn>\n\t\t</msup>\n\t</mrow>\n\t<mo>-</mo>\n\t<mrow>\n\t\t<mn>54</mn>\n\t\t<msup>\n\t\t\t<mi>y</mi>\n\t\t\t<mn>4</mn>\n\t\t</msup>\n\t</mrow>\n</mrow>\n</math> is equivalent to <math alttext=\"r y Superscript 4 Baseline left parenthesis 15 y minus 9 right parenthesis\"><mi>r</mi><msup><mi>y</mi><mn>4</mn></msup><mfenced><mrow><mrow><mn>15</mn></mrow><mi>y</mi><mo>-</mo><mrow><mn>9</mn></mrow></mrow></mfenced></math>, where <math alttext=\"r\"><mi>r</mi>\n</math> is a constant. What is the value of <math alttext=\"r\"><mi>r</mi>\n</math>?</p>","rationale_html":"<p>The correct answer is <math alttext=\"6\"><mn>6</mn>\n</math>. Applying the distributive property to the expression&nbsp;<math alttext=\"r y Superscript 4 Baseline left parenthesis 15 y minus 9 right parenthesis\"><mi>r</mi><msup><mi>y</mi><mn>4</mn></msup><mfenced><mrow><mn>15</mn><mi>y</mi><mo>-</mo><mn>9</mn></mrow></mfenced></math> yields <math alttext=\"15 r y Superscript 5 minus 9 r y Superscript 4\"><mn>15</mn><mi>r</mi><msup><mi>y</mi><mn>5</mn></msup><mo>-</mo><mn>9</mn><mi>r</mi><msup><mi>y</mi><mn>4</mn></msup></math>. Since <math alttext=\"90 y Superscript 5 minus 54 y Superscript 4\"><mn>90</mn><msup><mi>y</mi><mn>5</mn></msup><mo>-</mo><mn>54</mn><msup><mi>y</mi><mn>4</mn></msup></math> is equivalent to <math alttext=\"r y Superscript 4 Baseline left parenthesis 15 y minus 9 right parenthesis\"><mi>r</mi><msup><mi>y</mi><mn>4</mn></msup><mfenced><mrow><mn>15</mn><mi>y</mi><mo>-</mo><mn>9</mn></mrow></mfenced></math>, it follows that <math alttext=\"90 y Superscript 5 minus 54 y Superscript 4\"><mn>90</mn><msup><mi>y</mi><mn>5</mn></msup><mo>-</mo><mn>54</mn><msup><mi>y</mi><mn>4</mn></msup></math> is also equivalent to <math alttext=\"15 r y Superscript 5 minus 9 r y Superscript 4\"><mn>15</mn><mi>r</mi><msup><mi>y</mi><mn>5</mn></msup><mo>-</mo><mn>9</mn><mi>r</mi><msup><mi>y</mi><mn>4</mn></msup></math>. Since these expressions are equivalent, it follows that corresponding coefficients are equivalent. Therefore, <math alttext=\"90 equals 15 r\"><mrow>\n\t<mn>90</mn>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mn>15</mn>\n\t\t<mi>r</mi>\n\t</mrow>\n</mrow>\n</math> and <math alttext=\"negative 54 equals minus 9 r\"><mrow>\n\t<mo>-</mo><mn>54</mn>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mo>-</mo>\n\t\t<mn>9</mn>\n\t\t<mi>r</mi>\n\t</mrow>\n</mrow>\n</math>. Solving either of these equations for <math alttext=\"r\"><mi>r</mi>\n</math> will yield the value of <math alttext=\"r\"><mi>r</mi>\n</math>. Dividing both sides of <math alttext=\"90 equals 15 r\"><mrow>\n\t<mn>90</mn>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mn>15</mn>\n\t\t<mi>r</mi>\n\t</mrow>\n</mrow>\n</math> by <math alttext=\"15\"><mn>15</mn>\n</math> yields <math alttext=\"6 equals r\"><mrow>\n\t<mn>6</mn>\n\t<mo>=</mo>\n\t<mi>r</mi>\n</mrow>\n</math>. Therefore, the value of <math alttext=\"r\"><mi>r</mi>\n</math> is <math alttext=\"6\"><mn>6</mn>\n</math>. &nbsp;</p>","correct_answer":["6"],"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":[],"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}}