Difference between revisions of "2002 AMC 10B Problems/Problem 21"

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== Problem ==
 
== Problem ==
  
  
21. 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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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?
  
 
'''(A) Andy  (B) Beth  (C) Carlos  (D) Andy and Carlos tie for first.  (E) All three tie.'''
 
'''(A) Andy  (B) Beth  (C) Carlos  (D) Andy and Carlos tie for first.  (E) All three tie.'''
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== Solution ==
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let ''l''=area of Beth's lawn
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'
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'b''=how fast Beth's lawn mower cuts
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If Beth's lawn's area is ''l'' and her lawn mower's speed is ''b'', it will take her ''l''/''b'' time to finish.
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If Andy's lawn hs twice as much area as Beth's lawn, Andy's lawn's area would be 2''l''.
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If Andy's lawn has three times as much area as Carlos' lawn, Carlos' lawn's area would be 2''l''/3.
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If Carlos' lawn mower cuts half as fast as Beth's mower, Carlos' mower's speed is ''b''/2.
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If Carlos' lawn mower cuts one third as fast as Andy's mower, Andy's mower's speed is 3''b''/2.
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Since Andy's lawn's area is 2''l'' and his lawn mower's speed is 3''b''/2, it will take him 2''l''/(3''b''/2) time to finish, or 4''l''/3''b''.
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Since Carlos' lawn's area is 2''l''/3 and his lawn mower's speed is'' b''/2, it will take him (2''l''/3)/(''b''/2) time to finish, or 4''l''/3''b''.
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''l''/''b'' < 4''l''/3''b'' = 4''l''/3''b''
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Beth's time < Andy's time = Carlos' time
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So, Beth finishes first. The answer is '''(B)'''

Revision as of 13:41, 1 January 2011

Problem

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?

(A) Andy (B) Beth (C) Carlos (D) Andy and Carlos tie for first. (E) All three tie.


Solution

let l=area of Beth's lawn ' 'b=how fast Beth's lawn mower cuts

If Beth's lawn's area is l and her lawn mower's speed is b, it will take her l/b time to finish.

If Andy's lawn hs twice as much area as Beth's lawn, Andy's lawn's area would be 2l.

If Andy's lawn has three times as much area as Carlos' lawn, Carlos' lawn's area would be 2l/3.

If Carlos' lawn mower cuts half as fast as Beth's mower, Carlos' mower's speed is b/2.

If Carlos' lawn mower cuts one third as fast as Andy's mower, Andy's mower's speed is 3b/2.

Since Andy's lawn's area is 2l and his lawn mower's speed is 3b/2, it will take him 2l/(3b/2) time to finish, or 4l/3b.

Since Carlos' lawn's area is 2l/3 and his lawn mower's speed is b/2, it will take him (2l/3)/(b/2) time to finish, or 4l/3b.

l/b < 4l/3b = 4l/3b

Beth's time < Andy's time = Carlos' time

So, Beth finishes first. The answer is (B)

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