AMC 8 · 2010 · #4

Grade 6 arithmetic

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

mean-median-mode-rangefraction-arithmetic identify-subproblems ↑ Prerequisites: mean-median-mode-rangemulti-digit-arithmetic

📏 Short solution 💡 3 insights

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Problem

Take the data set 2, 3, 0, 3, 1, 4, 0, 3. Compute its mean, median, and mode, then add the three results together.

Pick an answer.

(A)

6.5

(B)

(C)

7.5

(D)

8.5

(E)

AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

The data is given in a jumbled order, so the first move is Tool #2 (Make a Systematic List): sort the eight numbers from smallest to largest. That single ordered list makes the median and mode obvious by inspection and keeps us from miscounting duplicates. Tool #7 (Identify Subproblems) is the natural frame for the whole problem because the question is really three small problems — find mean, find median, find mode — followed by one easy addition.

1STEP 1

Sort the eight numbers from smallest to largest so repeats and middle values stand out.

2, 3, 0, 3, 1, 4, 0, 3 → 0, 0, 1, 2, 3, 3, 3, 4

2STEP 2

One value repeats most in the sorted list, making the mode 3.

mode = 3

3STEP 3

With eight values, average the 4th and 5th sorted terms to get the median 2.5.

median =

\frac{2 + 3}{2}

= 2.5

4STEP 4

Add every value and divide by eight to get the mean 2.

mean =

\frac{0+0+1+2+3+3+3+4}{8}

\frac{16}{8}

= 2

5STEP 5

Add the three centers — mean, median, and mode — to reach the final answer.

mean + median + mode = 2 + 2.5 + 3 = 7.5 → (C)

Answer

7.5

The eight numbers all sit between 0 and 4, so each measure of center has to land in that range. We got mean = 2, median = 2.5, mode = 3 — all in [0, 4] and clustered where the data piles up (three 3s pull the mode and median upward, the two 0s drag the mean down). Their sum 7.5 lies between 0 + 0 + 0 = 0 and 4 + 4 + 4 = 12, and matches choice (C).

💡Key takeaway

This AMC 8 problem only needs Grade 6 measures of center — mean, median, and mode — you already know!

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