Difference between revisions of "2011 AMC 8 Problems/Problem 21"

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==Problem==
 
Students guess that Norb's age is <math>24, 28, 30, 32, 36, 38, 41, 44, 47</math>, and <math>49</math>. Norb says, "At least half of you guessed too low, two of you are off by one, and my age is a prime number." How old is Norb?
 
Students guess that Norb's age is <math>24, 28, 30, 32, 36, 38, 41, 44, 47</math>, and <math>49</math>. Norb says, "At least half of you guessed too low, two of you are off by one, and my age is a prime number." How old is Norb?
  
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==Solution==
 
==Solution==
If at least half the guesses are two low, then his age must be greater than <math>36.</math>
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If at least half the guesses are too low, then Norb's age must be greater than <math>36.</math>
  
If two of the guesses are off by one, then his age is in between two guesses whose difference is <math>2.</math> It could <math>31,37,</math> or <math>48,</math> but because it is greater than <math>36</math> it can only be <math>37</math> or <math>48.</math>
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If two of the guesses are off by one, then his age is in between two guesses whose difference is <math>2.</math> It could be <math>31,37,</math> or <math>48,</math> but because his age is greater than <math>36</math> it can only be <math>37</math> or <math>48.</math>
  
Lastly, his age is a prime number so the answer must be <math>\boxed{\textbf{(C)}\ 37}</math>
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Lastly, Norb's age is a prime number so the answer must be <math>\boxed{\textbf{(C)}\ 37}</math>
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2011|num-b=20|num-a=22}}
 
{{AMC8 box|year=2011|num-b=20|num-a=22}}
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{{MAA Notice}}

Revision as of 00:08, 10 November 2018

Problem

Students guess that Norb's age is $24, 28, 30, 32, 36, 38, 41, 44, 47$, and $49$. Norb says, "At least half of you guessed too low, two of you are off by one, and my age is a prime number." How old is Norb?

$\textbf{(A) }29\qquad\textbf{(B) }31\qquad\textbf{(C) }37\qquad\textbf{(D) }43\qquad\textbf{(E) }48$

Solution

If at least half the guesses are too low, then Norb's age must be greater than $36.$

If two of the guesses are off by one, then his age is in between two guesses whose difference is $2.$ It could be $31,37,$ or $48,$ but because his age is greater than $36$ it can only be $37$ or $48.$

Lastly, Norb's age is a prime number so the answer must be $\boxed{\textbf{(C)}\ 37}$

See Also

2011 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 20
Followed by
Problem 22
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AJHSME/AMC 8 Problems and Solutions

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