AMC 8 · 2016 · #6

Grade 6 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

mean-median-mode-rangegraph-reading systematic-enumerationidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticgraph-reading

📏 Short solution 💡 2 insights 📊 Diagram

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Problem

A bar graph shows how many of 19 people have a name of each length from 3 to 7 letters. The bars give the counts 7, 3, 1, 4, 4 for lengths 3, 4, 5, 6, 7. We need the median name length — the middle value once all 19 name lengths are lined up in order.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

The data come from a picture, so Tool #1 (Draw a Diagram) first — read each bar and write down the count per name length. Then Tool #7 (Identify Subproblems) splits the task into three small pieces: (a) confirm the total is 19, (b) find which position is the median, (c) walk the counts until we reach that position. Tool #2 (Make a Systematic List) is the workhorse: build a cumulative tally — "through length 3 we have 7, through length 4 we have 10, ..." — and the median position drops out by inspection. No algebra needed.

1STEP 1

Read the bar graph: each bar's height is the count for that name length, giving heights 7, 3, 1, 4, 4.

length 3 → 7, 4 → 3, 5 → 1, 6 → 4, 7 → 4

2STEP 2

Add the five bar heights to confirm the total is 19 people, matching the problem.

7 + 3 + 1 + 4 + 4 = 19

3STEP 3

With 19 values (odd), the median is the single middle one: the (19+1)/2 = 10th value in order.

(19+1)/2 = 10 → median = 10th value in order

4STEP 4

Cumulative tally: length-3 fills positions 1 to 7, then length-4 fills positions 8, 9, 10 — so the 10th value is in the length-4 group.

through length 3 & positions 1-7 ; through length 4 & positions 8-10 ; through length 5 & position 11 ; through length 6 & positions 12-15 ; through length 7 & positions 16-19

5STEP 5

The 10th value lands in the length-4 block, so the median name length is 4.

median = 4 → (B)

Answer

The smallest length (3) already accounts for 7 of 19 people — more than a third — so the median should sit near the low end. Length 4 is the very next category, which pushes the cumulative count from 7 to 10. Since 10 is exactly the median position for n = 19, the median has to be 4. A median of 5 or 6 would require fewer short names at the front, which the graph contradicts.

💡Key takeaway

Even with 19 names, you never have to write them all out — a short cumulative-count list shows the 10th name lands in the length-4 group.

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