2002 AMC 10B Problem 21

Problem 21 of 25HarderAlgebra

Andy’s lawn has twice as much area as Beth’s lawn and three times as much area as Carlos’ lawn. Carlos’ lawn mower cuts half as fast as Beth’s mower and one third as fast as Andy’s mower. If they all start to mow their lawns at the same time, who will finish first?

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Solution

Let Andy’s lawn have area

A,

so Beth’s is

\dfrac{A}{2}

and Carlos’ is

\dfrac{A}{3}.

Let Carlos mow at rate

R,

so Beth mows at

2R

and Andy at

3R.

The times are

\begin{aligned} &\text{Andy: } \dfrac{A}{3R}, \\ &\text{Beth: } \dfrac{\frac{A}{2}}{2R} = \dfrac{A}{4R}, \\ &\text{Carlos: } \dfrac{\frac{A}{3}}{R} = \dfrac{A}{3R}. \end{aligned}

Since

\dfrac{A}{4R}

is the smallest, Beth finishes first. Thus, the correct answer is B.

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