# 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 $32$, $36$ and $38$ and $47$ and $49$. 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 $\boxed{\textbf{(C)}\ 37}$.

## Video Solution by OmegaLearn

~ pi_is_3.14159265

## Video Solution 2

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

## Video Solution by WhyMath

~savannahsolver

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