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{ "quality": "blurry", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Solving Rational Equations", "Cross Multiplication" ], "new_concepts": [ "Rate-Time-Distance Problems", "Wind Speed Modeling" ], "current_concepts": [ "Solving Rational Equations", "Cross Multiplication", "Rate-Time-Distance Problems", "Wind Speed Modeling" ] } ## Define variables and expressions Let $w$ represent the speed of the wind in miles per hour (mph). When riding against the wind, the effective speed is: $$v_{\text{against}} = 20 - w$$ When riding with the wind, the effective speed is: $$v_{\text{with}} = 20 + w$$ ## Set up the rational equation Using the *Solving Rational Equations* knowledge point $$ \begin{aligned} t_{\text{against}} &= t_{\text{with}} \ \frac{2}{20 - w} &= \frac{3}{20 + w} \end{aligned} $$ ## Solve for the wind speed Using the *Cross Multiplication* knowledge point $$ \begin{aligned} 2(20 + w) &= 3(20 - w) \ 40 + 2w &= 60 - 3w \ 5w &= 20 \ w &= 4 \end{aligned} $$

(A) 4 mph

(B) 7 mph

(C) 10 mph

(D) 12 mph

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Solving Rational Equations" ] }

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Question

Explanation:

Define variables and expressions

Set up the rational equation

Solve for the wind speed

Answer: