2002 AMC 12B Problem 17

Problem 17 of 25IntermediateAlgebra

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;

then Beth’s is

\dfrac A2

and Carlos’ is

\dfrac A3.

With Carlos’ rate

R,

Beth mows at

2R

and Andy at

3R.

Their times are

\dfrac{A}{3R},

\dfrac{\frac{A}{2}}{2R}=\dfrac{A}{4R},

and

\dfrac{\frac{A}{3}}{R}=\dfrac{A}{3R}

respectively. Beth’s time

\dfrac{A}{4R}

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

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