Difference between revisions of "2025 AMC 8 Problems/Problem 9"
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==Problem== | ==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? | 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? | ||
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| + | [[File:Amc8_2025_prob9.PNG]] | ||
<math>\textbf{(A)}\ 5\qquad \textbf{(B)}\ 6.5\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9.5 \qquad \textbf{(E)}\ 12</math> | <math>\textbf{(A)}\ 5\qquad \textbf{(B)}\ 6.5\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9.5 \qquad \textbf{(E)}\ 12</math> | ||
==Solution 1== | ==Solution 1== | ||
| − | + | The answer can be expressed as (1+7)/2 + (2 + 8)/2 + ... + (6 + 12)/2, with the whole result divided by 6. Therefore, the answer of the question is the sum of the numbers from 1 through 12 divided by 2 * 6 = 12. The answer is 12/2 = 6 * 13 = 78/12, leading to the answer <math>\frac{78}{12}=\boxed{\textbf{(B)}~6.5}</math>. | |
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| + | ~Sigmacuber | ||
==Solution 2== | ==Solution 2== | ||
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~Bepin999 | ~Bepin999 | ||
| − | == | + | ==Video Solution 1 by SpreadTheMathLove== |
https://www.youtube.com/watch?v=jTTcscvcQmI | https://www.youtube.com/watch?v=jTTcscvcQmI | ||
==Video Solution by Thinking Feet== | ==Video Solution by Thinking Feet== | ||
https://youtu.be/PKMpTS6b988 | https://youtu.be/PKMpTS6b988 | ||
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| + | ==See Also== | ||
| + | {{AMC8 box|year=2025|num-b=8|num-a=10}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 00:21, 31 January 2025
Contents
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
The answer can be expressed as (1+7)/2 + (2 + 8)/2 + ... + (6 + 12)/2, with the whole result divided by 6. Therefore, the answer of the question is the sum of the numbers from 1 through 12 divided by 2 * 6 = 12. The answer is 12/2 = 6 * 13 = 78/12, leading to the answer 
.
~Sigmacuber
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
Video Solution 1 by SpreadTheMathLove
https://www.youtube.com/watch?v=jTTcscvcQmI
Video Solution by Thinking Feet
See Also
| 2025 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
