a can do a certain work in the same period in which b and c together can do it. if a and b together could do it in 10 days and c alone in 50 days, then in how many days could b alone do it?

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Work Rate Problems", "Systems of Equations" ], "current_concepts": [ "Work Rate Problems", "Systems of Equations" ] } ## Set up the rate equations $$ \begin{aligned} &R_A = R_B + R_C \ &R_A + R_B = \frac{1}{10} \ &R_C = \frac{1}{50} \end{aligned} $$ ## Solve for the individual rates $$ \begin{aligned} &(R_B + R_C) + R_B = \frac{1}{10} \ &2R_B + \frac{1}{50} = \frac{1}{10} \ &2R_B = \frac{5}{50} - \frac{1}{50} = \frac{4}{50} \ &R_B = \frac{2}{50} = \frac{1}{25} \end{aligned} $$ ## Calculate the time for B alone $$ T_B = \frac{1}{R_B} = 25 $$ 25 days { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Work Rate Problems" ] }

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Question

Explanation:

Set up the rate equations

Solve for the individual rates

Calculate the time for B alone

Answer: