2011 AMC 8 Problems/Problem 21

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 $\boxed{\textbf{(C)}\ 37}$

Soultion 2 (Alternative approach)

Since two guesses are off by one, we know that both $x+1$ and $x-1$ are in the list where $x$ is the age of Norb. Now, we know that $x+1$ and $x-1$ are $28$ and $30$ , $30$ and $32$ , and $36$ and $38$ . From these values, we know that $x$ must be $29$ , $31$ , and $37$ . Since half of the guesses are too low, $24, 28, 30, 32,$ and $36$ are all too low so we can eliminate all numbers in our list lesser than or equal to $36$ . Therefore, our list has only $37$ left so the answer is $\boxed{\textbf{(C)}\ 37}$ . ~ rabbit317

Video Solution by OmegaLearn

https://youtu.be/HISL2-N5NVg?t=3886

~ pi_is_3.14

Video Solution 2

https://youtu.be/lhBDgiYKpgs. Soo, DRMS, NM

Video Solution by WhyMath

https://youtu.be/i6BmM5W_Vm8

~savannahsolver

2011 AMC 8 (Problems • Answer Key • Resources)
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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