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Difference between revisions of "2025 AMC 8 Problems/Problem 9"

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==Solution 1==
 
==Solution 1==
 
Notice that this is simply the sum of the numbers on the clock face divided by <math>12</math>. Why? This is because for a given number, it will be divided by <math>2</math> due to its averaging with its counterpart on the other side of the clock face, and then it will be divided by <math>6</math> when we average out all the smaller averages. Since the sum of the first <math>12</math> positive integers is <math>78</math>, our answer is <math>\frac{78}{12}=\boxed{\textbf{(B)}~6.5}</math>. ~cxsmi
 
Notice that this is simply the sum of the numbers on the clock face divided by <math>12</math>. Why? This is because for a given number, it will be divided by <math>2</math> due to its averaging with its counterpart on the other side of the clock face, and then it will be divided by <math>6</math> when we average out all the smaller averages. Since the sum of the first <math>12</math> positive integers is <math>78</math>, our answer is <math>\frac{78}{12}=\boxed{\textbf{(B)}~6.5}</math>. ~cxsmi
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==Solution 2==
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The pairs for the opposite times are <math>(12,6)</math>, <math>(1,7)</math>, <math>(2,8)</math>, <math>(3,9)</math>, <math>(4,10)</math>, and <math>(5,11). The averages of each of these pairs are </math>9, 4, 5, 6, 7, 8<math> respectively. The averages of </math>9, 4, 5, 6, 7, 8<math> are </math>\frac{39}{6}=\boxed{\textbf{(B)}~6.5}$
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~Bepin999

Revision as of 22:12, 29 January 2025

Problem

Ningli looks at the 6 pairs of numbers directly across from each other on a clock. She takes the average of each pair of numbers. What is the average of the resulting 6 numbers?

$\textbf{(A)}\ 5\qquad \textbf{(B)}\ 6.5\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9.5 \qquad \textbf{(E)}\ 12$

Solution 1

Notice that this is simply the sum of the numbers on the clock face divided by $12$. Why? This is because for a given number, it will be divided by $2$ due to its averaging with its counterpart on the other side of the clock face, and then it will be divided by $6$ when we average out all the smaller averages. Since the sum of the first $12$ positive integers is $78$, our answer is $\frac{78}{12}=\boxed{\textbf{(B)}~6.5}$. ~cxsmi

Solution 2

The pairs for the opposite times are $(12,6)$, $(1,7)$, $(2,8)$, $(3,9)$, $(4,10)$, and $(5,11). The averages of each of these pairs are$9, 4, 5, 6, 7, 8$respectively. The averages of$9, 4, 5, 6, 7, 8$are$\frac{39}{6}=\boxed{\textbf{(B)}~6.5}$

~Bepin999

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