AMC 8 · 2025 · #9

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

mean-median-mode-rangesequences-arithmeticpattern-recognition pattern-recognitionsystematic-enumeration ↑ Prerequisites: fraction-arithmeticmean-median-mode-range

📏 Medium solution 💡 2 insights 📊 Diagram

Problem

On a standard clock face with numbers 1 through 12, Ningli pairs up the six numbers that sit directly opposite each other (a number and the one 6 hours across from it). For each of those 6 pairs she computes the average of the two numbers, getting 6 new numbers. We need the average of those 6 averages.

Pick an answer.

(A)

(B)

6.5

(C)

(D)

9.5

(E)

AMC 8 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

There are only 6 pairs, so the most direct attack is Tool #2 (Systematic List): list the pairs in order (1,7), (2,8), …, (6,12), compute each average, and then average those 6 numbers. While doing the list we can use Tool #5 (Pattern): the six pair-averages turn out to be 4, 5, 6, 7, 8, 9 — a perfectly regular arithmetic run, whose average is just the midpoint $\frac{(4+9)}{2}$ . Tool #1 (Diagram) shows why every pair must sum to 1+7 = 13, 2+8 = 10, 3+9 = 12 … wait, they do NOT all share the same sum, but they do all share the same average of $\frac{(a+b)}{2}$ that creeps up by 1 each time — exactly the pattern we exploit.

1STEP 1

Pair each clock number with the one across — 'opposite' means 6 positions apart, so k pairs with k+6.

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

2STEP 2

Average each pair with $\frac{(a+b)}{2}$ , which gives the list 4, 5, 6, 7, 8, 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,

\frac{(6+12)}{2}

3STEP 3

These six averages are consecutive integers, so their mean is just the midpoint — the average of the first and last.

Average of 4,5,6,7,8,9 =

\frac{(4+9)}{2}

\frac{13}{2}

= 6.5

4STEP 4

Confirm by adding all six values to get 39, then dividing by 6 to reach the same value.

\frac{(4+5+6+7+8+9)}{6}

\frac{39}{6}

\frac{13}{2}

= 6.5 → (B)

Answer

6.5

The clock numbers 1 through 12 have overall average

\frac{(1+2+…+12)}{12}

\frac{78}{12}

= 6.5. Since every clock number appears in exactly one pair and an average

\frac{(a+b)}{2}

weights each of its two members equally, averaging the six pair-averages is the same as averaging all twelve numbers. So the answer must be 6.5 — matching (B). Magnitude is plausible: 6.5 sits right between 1 and 12, exactly where a balanced average should land.

💡Key takeaway

This AMC 8 problem only needs Grade 6 "average is the center of the data" reasoning you already know — and once you see the pair-averages line up as 4, 5, 6, 7, 8, 9, the answer is just the middle: 6.5!

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