Difference between revisions of "2025 AMC 8 Problems/Problem 9"
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==Solution 2== | ==Solution 2== | ||
| − | The pairs for the opposite numbers 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)</math>. The averages of each of these pairs are <math>9, 4, 5, 6, 7,</math> and <math>8</math> respectively. The averages of <math>9, 4, 5, 6, 7,</math> and <math>8</math> are <math>\frac{39}{6}=\boxed{\textbf{(B)}~6.5}</math> | + | The pairs for the opposite numbers on the clock 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)</math>. The averages of each of these pairs are <math>9, 4, 5, 6, 7,</math> and <math>8</math> respectively. The averages of <math>9, 4, 5, 6, 7,</math> and <math>8</math> are <math>\frac{39}{6}=\boxed{\textbf{(B)}~6.5}</math> |
~Bepin999 | ~Bepin999 | ||
Revision as of 22:15, 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?
Solution 1
Notice that this is simply the sum of the numbers on the clock face divided by 
. Why? This is because for a given number, it will be divided by 
due to its averaging with its counterpart on the other side of the clock face, and then it will be divided by 
when we average out all the smaller averages. Since the sum of the first positive integers is 
, our answer is 
. ~cxsmi
Solution 2
The pairs for the opposite numbers on the clock are 
, 
, 
, 
, 
, and 
. The averages of each of these pairs are 
and 
respectively. The averages of and are 
~Bepin999
