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

Latest revision as of 23:28, 12 October 2025

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 1

At least half the guesses are too low, so 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 $\boxed{\textbf{(C)}\ 37}$ .

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

Solution 3 (Using the Answer Choices)

If his age is $29$ , then there won't be half of them who guessed less than his age. If his age is $31$ , then there still won't be half of them who guessed less than his age. If his age is $37$ , then there are half of them who guessed too low, half of them who guessed too high, and there are two people who guessed off by one. So, our answer is probably $\textbf{(C)}\ 37$ , but let's still check the other two choices. If his age is 43, then there won't be two people who were off by one. Finally, if his age is 48, then there won't be half of them who guessed too high. Therefore, our answer is $\boxed{\textbf{(C)}\ 37}$ .

~Yuhao2012

Video Solution by OmegaLearn

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

~ pi_is_3.14159265

Video Solution by WhyMath

https://youtu.be/i6BmM5W_Vm8

~savannahsolver

@@ Line 15: / Line 15: @@
 ==Solution 3 (Using the Answer Choices)==
-If his age is <math>29</math>, then there won't be half of them who guessed less than his age. If his age is <math>31</math>, then there still won't be half of them who guessed less than his age. If his age is <math>37</math>, then there are half of them who guessed too low, half of them who guessed too high, and there are two people who guessed off by one. So, our answer is probably <math>\boxed{\textbf{(C)}\ 37}</math>, but let's still check the other two choices. If his age is 43, then there won't be two people who were off by one. Finally, if his age is 48, then there won't be half of them who guessed too high. Therefore, our answer is <math>\boxed{\textbf{(C)}\ 37}</math>.
+If his age is <math>29</math>, then there won't be half of them who guessed less than his age. If his age is <math>31</math>, then there still won't be half of them who guessed less than his age. If his age is <math>37</math>, then there are half of them who guessed too low, half of them who guessed too high, and there are two people who guessed off by one. So, our answer is probably <math>\textbf{(C)}\ 37</math>, but let's still check the other two choices. If his age is 43, then there won't be two people who were off by one. Finally, if his age is 48, then there won't be half of them who guessed too high. Therefore, our answer is <math>\boxed{\textbf{(C)}\ 37}</math>.
 ~Yuhao2012

2011 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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