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

Latest revision as of 00:21, 31 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

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 $\frac{78}{12}=\boxed{\textbf{(B)}~6.5}$ .

~Sigmacuber

Solution 2

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

~Bepin999

Video Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=jTTcscvcQmI

Video Solution by Thinking Feet

https://youtu.be/PKMpTS6b988

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.

@@ Line 1: / Line 1: @@
 ==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?
+[[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>
 ==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
+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>.
+~Sigmacuber
 ==Solution 2==

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