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2025 AMC 8 Problems/Problem 9: Difference between revisions

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==Solution 1==
==Solution 1==
Notice that this is simply the sum of the numbers on the clock face, all 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

Revision as of 20:56, 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

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