AMC 8 · 2019 · #10

Grade 6 arithmetic

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

mean-median-mode-rangegraph-reading identify-subproblems ↑ Prerequisites: mean-median-mode-rangegraph-reading

📏 Medium solution 💡 2 insights 📊 Diagram

Problem

A bar chart shows the number of students at soccer practice for each of the five weekdays: Monday 20, Tuesday 26, Wednesday 16, Thursday 22, Friday 16. The mean and median were computed from these five values, but then Coach realizes Wednesday's count was actually 21, not 16. Which option correctly describes how the mean and the median each change after fixing that single Wednesday value?

Pick an answer.

(A)

The mean increases by 1 and the median does not change.

(B)

The mean increases by 1 and the median increases by 1.

(C)

The mean increases by 1 and the median increases by 5.

(D)

The mean increases by 5 and the median increases by 1.

(E)

The mean increases by 5 and the median increases by 5.

AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The question bundles two independent sub-questions: "what happens to the mean?" and "what happens to the median?". Tool #7 (Identify Subproblems) lets us answer each in isolation. For the mean, we use a shortcut: the sum increases by 21 - 16 = 5, spread over 5 data points, so the mean rises by 5 ÷ 5 = 1. For the median, Tool #15 (Organize Information in More Ways) tells us to re-sort the list (a different ordering of the same data) so the 3rd value pops out — once before the fix, once after. Tool #3 (Eliminate Possibilities) is the multiple-choice cleanup: once we know both changes are +1, only choice (B) survives.

1STEP 1

Read the five values straight off the bar chart: Monday 20, Tuesday 26, Wednesday 16, Thursday 22, Friday 16.

{20, 26, 16, 22, 16}

2STEP 2

Sort the values smallest to largest; with five numbers the middle one is the 3rd, so the original median is 20.

sorted: {16, 16, 20, 22, 26} → median_old = 20

3STEP 3

Add the five values and divide by 5: 100 ÷ 5 gives an original mean of 20.

\frac{20+26+16+22+16}{5}

\frac{100}{5}

= 20 → mean_old = 20

4STEP 4

Swap Wednesday's 16 for 21 and re-sort to {16, 20, 21, 22, 26}: the new middle is 21, so the median rises by 1.

new sorted: {16, 20, 21, 22, 26} → median_new = 21, Δmedian = +1

5STEP 5

Wednesday's value climbs by 5, so the sum climbs by 5; over 5 data points the mean rises by 1 (new mean 105 ÷ 5 = 21).

Δmean =

\frac{Δsum}{n}

= +

\frac{5}{5}

= +1

6STEP 6

Both statistics rise by 1, which matches only option (B); (A), (C), (D), (E) each miss one of the two changes.

Δmean = +1, Δmedian = +1 → (B)

Answer

The mean increases by 1 and the median increases by 1.

Bumping just one of five values up by 5 should pull the mean up by exactly 5 / 5 = 1, which we got. The change moved that value from 16 (a low end) past the old middle 20 to 21, so the sorted order shifts and the new middle reads 21 — also +1. Both shifts are in the same direction (the data set got slightly bigger overall) and modest in size, consistent with changing only one of five points.

💡Key takeaway

This AMC 8 problem only needs Grade 6 mean and median you already know — plus the handy shortcut that when one value goes up by 5, the mean of 5 numbers goes up by just 1!

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