2025 AMC 8 Problem 9

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To help you master 2025 AMC 8 Problem 9, this page includes:

• The original AMC problem to try, plus the correct answer

• Clear, step-by-step solutions using different methods

• Key ideas, useful formulas, and takeaways you can apply to similar problems

• Related problems from past AMC exams, plus new practice problems that build the same skills

Want more practice? Scroll down to download Think Academy’s AMC 8 practice packs and keep building your AMC 8 skills.

2025 AMC8 Problem 9

Solution

Solution 1

Direct Calculation Method

Identify every pair of opposite numbers on the clock, compute the average of each pair, and then calculate the final average of these six results.

The average (arithmetic mean) of a set of numbers is found by adding all the numbers together and dividing by the total count of the numbers.

Step 1: Identify the pairs of opposite numbers

Look at the clock face. The number 12 is at the top and 6 is at the bottom, forming our first pair. Continuing around the clock, each number is paired with the one directly across from it. We can list all 6 pairs.

$$(12,6),(1,7),(2,8),(3,9),(4,10),(5,11)$$

Step 2: Calculate the average of each pair

The average of two numbers is their sum divided by 2. We calculate this simple average for each of the 6 pairs.

$$\frac{12+6}{2}=9$$

$$\frac{1+7}{2}=4$$

$$\frac{2+8}{2}=5$$

$$\frac{3+9}{2}=6$$

$$\frac{4+10}{2}=7$$

$$\frac{5+11}{2}=8$$

Step 3: Sum the resulting averages

Ningli takes the 6 numbers that resulted from the previous step. To find their average, we must first add them all together.

$$9+4+5+6+7+8=39$$

Step 4: Calculate the final average

Because there are 6 numbers in total, we divide their sum by 6 to find the final average.

$$\frac{39}{6}=6.5$$

Correct answer: B. $6.5$
The final average is 6.5, which corresponds to choice B. We can verify this by checking that the 6 averages (4, 5, 6, 7, 8, 9) are symmetrically distributed around 6.5.

Solution 2

Algebraic Pattern Method

Represent each pair algebraically as a variable and a constant. This reveals that the six averages form a simple, evenly spaced list of consecutive integers.

The average of any evenly spaced sequence (an arithmetic sequence) is exactly halfway between its smallest and largest values. You do not need to sum all the terms.

Step 1: Identify the algebraic relationship between opposite numbers

For any number on the first half of the clock (from 1 to 6), the number directly across from it is exactly 6 hours ahead. If we call the first number in the pair x, the opposite number is x + 6.

$$\text{Pair:}(x,x+6)$$

Step 2: Find a general expression for the average of a pair

We can add the two numbers in our algebraic pair and divide by 2 to find a general formula for their average.

$$\frac{x+(x+6)}{2}=\frac{2x+6}{2}=x+3$$

Step 3: Determine the sequence of the six averages

The first number in each pair, x, takes the consecutive integer values 1 through 6. Substituting these into our simplified expression shows that the 6 resulting averages are just the numbers 4 through 9.

$$1+3=4$$

$$2+3=5$$

$$3+3=6$$

$$4+3=7$$

$$5+3=8$$

$$6+3=9$$

Step 4: Calculate the average of the consecutive sequence

The resulting averages form a sequence of consecutive integers. The average of such a sequence is simply the average of its first and last terms.

$$\frac{4+9}{2}=6.5$$

Correct answer: B. $6.5$
The final average is 6.5 (Choice B). The sequence 4, 5, 6, 7, 8, 9 is perfectly symmetric, and its two middle numbers are 6 and 7. Halfway between them is 6.5, confirming our calculation.

Solution 3

Global Average Method

The average of the 6 pair-averages is exactly the same as the average of all 12 numbers on the clock, because every number is counted exactly once with the same weight.

If a dataset is partitioned into equal-sized groups, the average of the group averages is exactly equal to the overall average of the entire dataset.

Step 1: Relate the group averages to the overall average

The problem asks for the average of 6 values, each of which is the average of 2 numbers. Because every number from 1 to 12 is used exactly once in these pairs, finding the average of these 6 pair-averages is mathematically identical to finding the average of all 12 numbers together.

$$\text{Final Average}=\frac{1}{6}\left(\frac{a_1+b_1}{2}+\cdots +\frac{a_6+b_6}{2}\right)$$

$$\text{Final Average}=\frac{a_1+b_1+\cdots +a_6+b_6}{12}$$

Step 2: Identify the complete set of numbers

The numbers on the clock are simply the consecutive integers from 1 to 12. Because they are evenly spaced, we do not need to add them all up to find their average.

$$1,2,3,\dots ,12$$

Step 3: Calculate the overall average using the first and last terms

We can find the average of this consecutive sequence by taking the average of just the smallest number (1) and the largest number (12).

$$\frac{1+12}{2}=\frac{13}{2}=6.5$$

Correct answer: B. $6.5$
The final average is 6.5, which corresponds to choice B. By completely bypassing the intermediate pairing step, we avoid potential arithmetic mistakes and arrive at the same logical result instantly.

Similar Problems in Real Exams

The following five AMC 8 problems are closely related to AMC 8 2025 Question 9. These real AMC 8 problems test the same key concept—Means, Averages & Weighted Values. Click any problem to learn and practice, then continue to the adapted practice section later in this article for more problems to try.

2022 AMC 8 Problem 21

2020 AMC 8 Problem 2

2019 AMC 8 Problem 7

2018 AMC 8 Problem 10

2018 AMC 8 Problem 13

Adapted Problems to Practice

What Formulas Do You Need for This Problem?

• Average of an Arithmetic Sequence:

Applies to any sequence where the difference between consecutive terms is constant (e.g., 1, 2, 3, 4 or 4, 6, 8, 10). Used to quickly find the average of the numbers 1 through 12, or the sequence 4 through 9, without adding all the terms together.

• Average of Equal-Sized Groups:

Applies only when every group contains the same number of items and every item from the total set is included exactly once. Each of the 6 pairs on the clock contains exactly 2 numbers, and together they cover all 12 numbers on the clock exactly once. This structural equality allows us to find the average of all 12 numbers directly.

Key Takeaway: Remember This Strategy for Next Time

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