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

Revision as of 16:25, 2 February 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 $12/2$ = $6 \times 13$ = $78/12$ , 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

Solution 3 (most efficient solution)

If you read the problem carefully, you will find that it is basically asking for the average of all $12$ numbers. To find the average of all $12$ numbers, you add all the numbers from $1 - 12$ which is $78$ , and divide it by $12$ because there are 12 terms, therefore, the answer is $\frac{78}{12}=\boxed{\textbf{(B)}~6.5}$ .

~JacQueen2024

Video Solution 1 by SpreadTheMathLove

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

Video Solution (A Clever Explanation You’ll Get Instantly)

https://youtu.be/VP7g-s8akMY?si=QWgBnVJLRi_J9Hox&t=667 ~hsnacademy

Video Solution by Thinking Feet

https://youtu.be/PKMpTS6b988

@@ Line 7: / Line 7: @@
 ==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>.
+The answer can be expressed as <math>(1+7)/2 + (2 + 8)/2 + ... + (6 + 12)/2</math>, 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 <math>12/2</math> = <math>6 \times 13</math> = <math>78/12</math>, leading to the answer <math>\frac{78}{12}=\boxed{\textbf{(B)}~6.5}</math>.
 ~Sigmacuber

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