{"id":"cb_question_bank:0c4cf983-0e31-473d-ba75-7107c2754540","source_id":"cb_question_bank","source_item_id":"0c4cf983-0e31-473d-ba75-7107c2754540","external_id":"555b2745-a82f-4231-a6df-c31085a80638","ibn":null,"question_id":"e5c57163","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.A.","skill":"Area and volume","difficulty":"H","score_band":6,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Square A has side lengths that are 166 times the side lengths of square B. The area of square A is k times the area of square B. What is the value of k ?","stimulus":"","stimulus_html":null,"stem_html":"<p>Square A has side lengths that are <math alttext=\"166\"><mn>166</mn>\n</math> times the side lengths of square B. The area of square A is <math alttext=\"k\"><mi>k</mi>\n</math> times the area of square B. What is the value of <math alttext=\"k\"><mi>k</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"27,556\"><mn>27,556</mn>\n</math>. The area of a square is <math alttext=\"s squared\"><msup>\n\t<mi>s</mi>\n\t<mn>2</mn>\n</msup>\n</math>, where <math alttext=\"s\"><mi>s</mi>\n</math> is the side length of the square. Let <math alttext=\"x\"><mi>x</mi>\n</math> represent the length of each side of square B. Substituting <math alttext=\"x\"><mi>x</mi>\n</math> for <math alttext=\"s\"><mi>s</mi>\n</math> in <math alttext=\"s squared\"><msup>\n\t<mi>s</mi>\n\t<mn>2</mn>\n</msup>\n</math> yields <math alttext=\"x squared\"><msup>\n\t<mi>x</mi>\n\t<mn>2</mn>\n</msup>\n</math>. It follows that the area of square B is <math alttext=\"x squared\"><msup>\n\t<mi>x</mi>\n\t<mn>2</mn>\n</msup>\n</math>. It’s given that square A has side lengths that are <math alttext=\"166\"><mn>166</mn>\n</math> times the side lengths of square B. Since <math alttext=\"x\"><mi>x</mi>\n</math> represents the length of each side of square B, the length of each side of square A can be represented by the expression <math alttext=\"166 x\"><mrow>\n\t<mn>166</mn>\n\t<mi>x</mi>\n</mrow>\n</math>. It follows that the area of square A is <math alttext=\"left parenthesis 166 x right parenthesis squared\"><msup><mfenced><mrow><mn>166</mn><mi>x</mi></mrow></mfenced><mn>2</mn></msup></math>, or <math alttext=\"27,556 x squared\"><mrow>\n\t<mn>27,556</mn>\n\t<msup>\n\t\t<mi>x</mi>\n\t\t<mn>2</mn>\n\t</msup>\n</mrow>\n</math>. It’s given that the area of square A is <math alttext=\"k\"><mi>k</mi>\n</math> times the area of square B. Since the area of square A is equal to <math alttext=\"27,556 x squared\"><mrow>\n\t<mn>27,556</mn>\n\t<msup>\n\t\t<mi>x</mi>\n\t\t<mn>2</mn>\n\t</msup>\n</mrow>\n</math>, and the area of square B is equal to <math alttext=\"x squared\"><msup>\n\t<mi>x</mi>\n\t<mn>2</mn>\n</msup>\n</math>, an equation representing the given statement is <math alttext=\"27,556 x squared equals k x squared\"><mrow>\n\t<mrow>\n\t\t<mn>27,556</mn>\n\t\t<msup>\n\t\t\t<mi>x</mi>\n\t\t\t<mn>2</mn>\n\t\t</msup>\n\t</mrow>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mi>k</mi>\n\t\t<msup>\n\t\t\t<mi>x</mi>\n\t\t\t<mn>2</mn>\n\t\t</msup>\n\t</mrow>\n</mrow>\n</math>. Since <math alttext=\"x\"><mi>x</mi>\n</math> represents the length of each side of square B, the value of <math alttext=\"x\"><mi>x</mi>\n</math> must be positive. Therefore, the value of <math alttext=\"x squared\"><msup>\n\t<mi>x</mi>\n\t<mn>2</mn>\n</msup>\n</math> is also positive, so it does not equal <math alttext=\"0\"><mn>0</mn>\n</math>. Dividing by <math alttext=\"x squared\"><msup>\n\t<mi>x</mi>\n\t<mn>2</mn>\n</msup>\n</math> on both sides of the equation <math alttext=\"27,556 x squared equals k x squared\"><mrow>\n\t<mrow>\n\t\t<mn>27,556</mn>\n\t\t<msup>\n\t\t\t<mi>x</mi>\n\t\t\t<mn>2</mn>\n\t\t</msup>\n\t</mrow>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mi>k</mi>\n\t\t<msup>\n\t\t\t<mi>x</mi>\n\t\t\t<mn>2</mn>\n\t\t</msup>\n\t</mrow>\n</mrow>\n</math> yields <math alttext=\"27,556 equals k\"><mn>27,556</mn><mo>=</mo><mi>k</mi></math>. Therefore, the value of <math alttext=\"k\"><mi>k</mi>\n</math> is <math alttext=\"27,556\"><mn>27,556</mn>\n</math>.</p>","correct_answer":["27556"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.837000+00:00","source_updated_at":"2023-08-02T20:25:59.837000+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}}