twin brothers, billy and bobby, can mow their grandparents lawn together in 75 minutes. billy could mow the lawn by himself in 15 minutes less time that it would take bobby. how long would it take bobby to mow the lawn by himself?

it would take bobby minutes to mow the lawn by himself.
(round to the nearest minute.)

Question

twin brothers, billy and bobby, can mow their grandparents lawn together in 75 minutes. billy could mow the lawn by himself in 15 minutes less time that it would take bobby. how long would it take bobby to mow the lawn by himself?

it would take bobby minutes to mow the lawn by himself.
(round to the nearest minute.)

it would take bobby   minutes to mow the lawn by himself.
(round to the nearest minute.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Work Rate Problems", "Rational Equations", "Quadratic Formula" ], "new_concepts": [], "current_concepts": [ "Work Rate Problems", "Rational Equations", "Quadratic Formula" ] } ## Set up the work rate equation $$ \begin{aligned} &\text{Let } x = \text{time for Bobby to mow the lawn alone (in minutes)} \ &\text{Billy's time} = x - 15 \ &\frac{1}{x} + \frac{1}{x - 15} = \frac{1}{75} \end{aligned} $$ ## Solve the rational equation for x $$ \begin{aligned} &75(x - 15) + 75x = x(x - 15) \ &75x - 1125 + 75x = x^2 - 15x \ &150x - 1125 = x^2 - 15x \ &x^2 - 165x + 1125 = 0 \end{aligned} $$ ## Apply the quadratic formula $$ \begin{aligned} &x = \frac{-(-165) \pm \sqrt{(-165)^2 - 4(1)(1125)}}{2(1)} \ &x = \frac{165 \pm \sqrt{27225 - 4500}}{2} \ &x = \frac{165 \pm \sqrt{22725}}{2} \ &x \approx \frac{165 \pm 150.748}{2} \ &x_1 \approx \frac{315.748}{2} \approx 157.87 \ &x_2 \approx \frac{14.252}{2} \approx 7.13 \quad (\text{rejected since } x > 15) \ &x \approx 158 \text{ minutes} \end{aligned} $$ Twin brothers, Billy and Bobby, can mow their grandparent's lawn together in 75 minutes. Billy could mow the lawn by himself in 15 minutes less time that it would take Bobby. How long would it take Bobby to mow the lawn by himself? It would take Bobby 158 minutes to mow the lawn by himself. (Round to the nearest minute.) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Work Rate Problems" ] }

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QUESTION IMAGE

Question

Explanation:

Set up the work rate equation

Solve the rational equation for x

Apply the quadratic formula

Answer: