AMC 8 · 2020 · #14

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

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

📏 Short solution 💡 2 insights 📊 Diagram

Problem

A bar chart shows the populations of all 20 cities in Newton County, and a horizontal dashed line marks the average population. Without summing the 20 individual bars, estimate the total population from the dashed line, and pick the closest of the five choices.

Pick an answer.

(A)

$65{,}000$

(B)

$75{,}000$

(C)

$85{,}000$

(D)

$95{,}000$

(E)

$105{,}000$

AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Work Backwards

The chart already gives us the answer to a later step in the average calculation: the average was found by taking the total and dividing by 20. So we just undo that division — Tool #11 (Work Backwards) — by multiplying the average back by 20. Tool #1 (Draw a Diagram) is used to read the dashed line's value off the y-axis carefully (the small tick marks between the labeled gridlines matter). Tool #3 (Eliminate Possibilities) is the natural AMC closer: once we compute a total, we pick the closest of the five round choices.

1STEP 1

Read the dashed line off the scaled y-axis: it sits halfway between the 4,500 and 5,000 ticks, so the average reads as 4,750.

average =

\frac{4500 + 5000}{2}

= 4750

2STEP 2

By definition the dashed average is the total divided by 20, so working backwards, total = average × 20.

average =

\frac{total population}{20}

⟹ total = average × 20

3STEP 3

Substitute 4,750 into that relation: 4,750 × 20 = 4,750 × 2 × 10 = 95,000.

total = 4750 × 20 = 9500 × 10 = 95,000

4STEP 4

The choices are spaced 10,000 apart and 95,000 is exactly one of them, so the closest option is (D).

|95,000 - 95,000| = 0 → (D)

Answer

95,000

Sanity-check by eyeballing the bars: most bars sit between 1,500 and 8,000, and an average around 4,750 looks visually right (about half the bars above the dashed line, half below). The total 20 × 4,750 = 95,000 also matches the bar-by-bar sum: 8750+3800+5000+2900+6400+7500+4100+1400+2600+1470+2600+7100+4070+7500+7000+8100+1900+1600+5850+5750 = 95,000 exactly. Answer (D) is consistent.

💡Key takeaway

This AMC 8 problem only needs Grade 6 "mean as a measure of center" — once you know average × count = total, you already know enough to solve it!

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