{"id":"cb_question_bank:9e34a770-e2e1-43d3-abf5-9379de7217b7","source_id":"cb_question_bank","source_item_id":"9e34a770-e2e1-43d3-abf5-9379de7217b7","external_id":"89816479-b500-4fe9-b80f-8f9fea007b9f","ibn":null,"question_id":"7e1bff94","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.B.","skill":"Linear functions","difficulty":"M","score_band":4,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The table gives the number of hours, h , of labor and a plumber’s total charge f left parenthesis h right parenthesis , in dollars, for two different jobs. h f left parenthesis h right parenthesis 1 155 3 285 There is a linear relationship between h and f left parenthesis h right parenthesis . Which equation represents this relationship?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">The table gives the number of hours, <math alttext=\"h\"><mi>h</mi>\n</math>, of labor and a plumber’s total charge <math alttext=\"f left parenthesis h right parenthesis\"><mi>f</mi><mfenced><mi>h</mi></mfenced></math>, in dollars, for two different jobs.</p>\n<figure class=\"table\"><table border=\"1\">\n<thead>\n<tr>\n<th style=\"width: 47.6236%; text-align: center; vertical-align: bottom;\" scope=\"col\"><math alttext=\"h\"><mi>h</mi>\n</math></th>\n<th style=\"width: 47.7844%; text-align: center; vertical-align: bottom;\" scope=\"col\"><math alttext=\"f left parenthesis h right parenthesis\"><mi>f</mi><mfenced><mi>h</mi></mfenced></math></th>\n</tr>\n</thead>\n<tbody>\n<tr>\n<td style=\"width: 47.6236%; text-align: center;\"><math alttext=\"1\"><mn>1</mn>\n</math></td>\n<td style=\"width: 47.7844%; text-align: center;\"><math alttext=\"155\"><mn>155</mn>\n</math></td>\n</tr>\n<tr>\n<td style=\"width: 47.6236%; text-align: center;\"><math alttext=\"3\"><mn>3</mn>\n</math></td>\n<td style=\"width: 47.7844%; text-align: center;\"><math alttext=\"285\"><mn>285</mn>\n</math></td>\n</tr>\n</tbody>\n</table></figure>\n<p style=\"text-align: left;\">There is a linear relationship between <math alttext=\"h\"><mi>h</mi>\n</math> and&nbsp;<math alttext=\"f left parenthesis h right parenthesis\"><mi>f</mi><mfenced><mi>h</mi></mfenced></math>. Which equation represents this relationship?</p>","rationale_html":"<p style=\"text-align: left;\">Choice C is correct. It's given that there is a linear relationship between a plumber's hours of labor, <math alttext=\"h\"><mi>h</mi>\n</math>, and the plumber's total charge <math alttext=\"f left parenthesis h right parenthesis\"><mi>f</mi><mfenced><mi>h</mi></mfenced></math>, in dollars. It follows that the relationship can be represented by an equation of the form <math alttext=\"f left parenthesis h right parenthesis equals m h plus b\"><mi>f</mi><mfenced><mi>h</mi></mfenced><mo>=</mo><mi>m</mi><mi>h</mi><mo>+</mo><mi>b</mi></math>, where <math alttext=\"m\"><mi>m</mi>\n</math> is the rate of change of the function <math alttext=\"f\"><mi>f</mi>\n</math> and <math alttext=\"b\"><mi>b</mi>\n</math> is a constant. The rate of change of <math alttext=\"f\"><mi>f</mi>\n</math> can be calculated by dividing the difference in two values of <math alttext=\"f left parenthesis h right parenthesis\"><mi>f</mi><mfenced><mi>h</mi></mfenced></math> by the difference in the corresponding values of <math alttext=\"h\"><mi>h</mi>\n</math>. Based on the values given in the table, the rate of change of <math alttext=\"f\"><mi>f</mi>\n</math> is&nbsp;<math alttext=\"StartFraction 285 minus 155 Over 3 minus 1 EndFraction\"><mfrac><mrow><mn>285</mn><mo>-</mo><mn>155</mn></mrow><mrow><mn>3</mn><mo>-</mo><mn>1</mn></mrow></mfrac></math>, or <math alttext=\"65\"><mn>65</mn>\n</math>. Substituting <math alttext=\"65\"><mn>65</mn>\n</math> for <math alttext=\"m\"><mi>m</mi>\n</math> in the equation <math alttext=\"f left parenthesis h right parenthesis equals m h plus b\"><mi>f</mi><mfenced><mi>h</mi></mfenced><mo>=</mo><mi>m</mi><mi>h</mi><mo>+</mo><mi>b</mi></math> yields <math alttext=\"f left parenthesis h right parenthesis equals 65 h plus b\"><mi>f</mi><mfenced><mi>h</mi></mfenced><mo>=</mo><mn>65</mn><mi>h</mi><mo>+</mo><mi>b</mi></math>. The value of <math alttext=\"b\"><mi>b</mi>\n</math> can be found by substituting any value of <math alttext=\"h\"><mi>h</mi>\n</math> and its corresponding value of <math alttext=\"f left parenthesis h right parenthesis\"><mi>f</mi><mfenced><mi>h</mi></mfenced></math> for <math alttext=\"h\"><mi>h</mi>\n</math> and&nbsp;<math alttext=\"f left parenthesis h right parenthesis\"><mi>f</mi><mfenced><mi>h</mi></mfenced></math>, respectively, in this equation. Substituting <math alttext=\"1\"><mn>1</mn>\n</math> for <math alttext=\"h\"><mi>h</mi>\n</math> and <math alttext=\"155\"><mn>155</mn>\n</math> for <math alttext=\"f left parenthesis h right parenthesis\"><mi>f</mi><mfenced><mi>h</mi></mfenced></math> yields <math alttext=\"155 equals 65 left parenthesis 1 right parenthesis plus b\"><mn>155</mn><mo>=</mo><mn>65</mn><mfenced><mn>1</mn></mfenced><mo>+</mo><mi>b</mi></math>, or <math alttext=\"155 equals 65 plus b\"><mn>155</mn><mo>=</mo><mn>65</mn><mo>+</mo><mi>b</mi></math>. Subtracting <math alttext=\"65\"><mn>65</mn>\n</math> from both sides of this equation yields <math alttext=\"90 equals b\"><mrow>\n\t<mn>90</mn>\n\t<mo>=</mo>\n\t<mi>b</mi>\n</mrow>\n</math>. Substituting <math alttext=\"90\"><mn>90</mn>\n</math> for <math alttext=\"b\"><mi>b</mi>\n</math> in the equation <math alttext=\"f left parenthesis h right parenthesis equals 65 h plus b\"><mi>f</mi><mfenced><mi>h</mi></mfenced><mo>=</mo><mn>65</mn><mi>h</mi><mo>+</mo><mi>b</mi></math> yields&nbsp;<math alttext=\"f left parenthesis h right parenthesis equals 65 h plus 90\"><mi>f</mi><mo>(</mo><mi>h</mi><mo>)</mo><mo>=</mo><mn>65</mn><mi>h</mi><mo>+</mo><mn>90</mn></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 B is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["C"],"has_media":false,"has_mathml":true,"has_table":true,"source_created_at":"2023-08-02T20:25:59.812000+00:00","source_updated_at":"2023-08-02T20:25:59.812000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"f left parenthesis h right parenthesis equals 25 h plus 130\"><mi>f</mi><mfenced><mi>h</mi></mfenced><mo>=</mo><mrow><mn>25</mn></mrow><mi>h</mi><mo>+</mo><mrow><mn>130</mn></mrow></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"f left parenthesis h right parenthesis equals 130 h plus 25\"><mi>f</mi><mfenced><mi>h</mi></mfenced><mo>=</mo><mrow><mn>130</mn></mrow><mi>h</mi><mo>+</mo><mrow><mn>25</mn></mrow></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"f left parenthesis h right parenthesis equals 65 h plus 90\"><mi>f</mi><mfenced><mi>h</mi></mfenced><mo>=</mo><mrow><mn>65</mn></mrow><mi>h</mi><mo>+</mo><mrow><mn>90</mn></mrow></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"f left parenthesis h right parenthesis equals 90 h plus 65\"><mi>f</mi><mfenced><mi>h</mi></mfenced><mo>=</mo><mrow><mn>90</mn></mrow><mi>h</mi><mo>+</mo><mrow><mn>65</mn></mrow></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}}