2002 AMC 12B Problem 17

Question

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?

Andy

Beth

Carlos

Andy and Carlos tie for first.

All three tie.

Answer: B

Concepts:rate ratio and proportion

Difficulty rating: 1370

Small hint:

Let Andy’s area be

A

; then the areas are

A,

\dfrac A2,

\dfrac A3

and the mowing rates are

3R,

2R,

R

Big hint:

Time equals area divided by rate; compare

\dfrac{A}{3R},

\dfrac{A}{4R},

\dfrac{A}{3R}

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.