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

There are $20$ cities in the County of Newton. Their populations are shown in the bar chart below. The average population of all the cities is indicated by the horizontal dashed line. Which of the following is closest to the total population of all $20$ cities?

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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: There are exactly $20$ cities; Each city's population is shown as a bar on the chart; A horizontal dashed line marks the average population of the $20$ cities; The y-axis is labeled at $2000, 4000, 6000, 8000$, with smaller tick marks every $500$; Answer choices: (A) $65{,}000$, (B) $75{,}000$, (C) $85{,}000$, (D) $95{,}000$, (E) $105{,}000$

Unknowns: The total population of all $20$ cities (the choice closest to it)

Understand

Plan

Primary tool: #11 Work Backwards

Secondary: #1 Draw a Diagram, #3 Eliminate Possibilities

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.

Execute — Answer: D

#1 Draw a Diagram 3.MD.B.3 Step 1

Read the average from the dashed line.
The y-axis has bold labels at $2000, 4000, 6000, 8000$, and the minor tick marks divide each $2000$ interval into four equal $500$ steps.
Between $4000$ and $6000$ the ticks fall at $4500, 5000, 5500$.
The dashed line sits exactly halfway between the $4{,}500$ and $5{,}000$ ticks.

$$\text{average} = \dfrac{4500 + 5000}{2} = 4750$$

💡 Reading a value off a bar-graph's scaled y-axis is exactly the Grade 3 scaled bar-graph skill.

#11 Work Backwards 6.SP.A.3 Step 2

Recall what "average" means.
The chart's dashed line was produced by adding up all $20$ city populations and then dividing by $20$.
So $\text{average} \times 20$ undoes the division and gives back the total — this is the Tool #11 "Work Backwards" move.

$$\text{average} = \dfrac{\text{total population}}{20} \;\Longrightarrow\; \text{total} = \text{average} \times 20$$

💡 Knowing that an average summarizes a whole data set with one number (so that number times the count gives back the sum) is the Grade 6 "measure of center" idea.

#11 Work Backwards 4.NBT.B.5 Step 3

Plug the average from Step 1 into the relation from Step 2.
Multiplying $4{,}750$ by $20$ is easiest as $4750 \times 2 \times 10$.

$$\text{total} = 4750 \times 20 = 9500 \times 10 = 95{,}000$$

💡 Multiplying a four-digit number by a one-digit number and then by $10$ is a Grade 4 multi-digit multiplication.

#3 Eliminate Possibilities 4.NBT.A.2 Step 4

Compare $95{,}000$ to the five choices.
The options are spaced $10{,}000$ apart, and our total $95{,}000$ is exactly one of them, so the "closest" option is itself.

$$|95{,}000 - 95{,}000| = 0 \;\Rightarrow\; \textbf{(D)}$$

💡 Comparing whole-number magnitudes to pick the nearest choice is a Grade 4 place-value comparison.

[1] #1 3.MD.B.3 Read the average from the dashed line. The y-axis has bold labels at $2000, 4000

[2] #11 6.SP.A.3 Recall what "average" means. The chart's dashed line was produced by adding up a

[3] #11 4.NBT.B.5 Plug the average from Step 1 into the relation from Step 2. Multiplying $4{,}750

[4] #3 4.NBT.A.2 Compare $95{,}000$ to the five choices. The options are spaced $10{,}000$ apart,

Review

Reasonableness: 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 $20 \times 4{,}750 = 95{,}000$ 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.

Alternative: Tool #7 (Identify Subproblems) plus direct addition: split the $20$ bars into two groups of $10$, sum each group, then add the two sums. This works but takes much longer and risks arithmetic slips — the average shortcut is the elegant AMC move because the test-maker has already done the summing work for you.

CCSS standards used (min grade 6)

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs (Reading the value $4{,}750$ off the y-axis where the dashed average line sits (halfway between the $4{,}500$ and $5{,}000$ ticks).)
6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number (Using the fact that the average $=$ (total population) $\div 20$, so the total population equals the average times $20$.)
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number (Computing $4{,}750 \times 20 = 95{,}000$ via $4750 \times 2 \times 10$.)
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols (Comparing $95{,}000$ to the five choices to pick the closest option (D).)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 6)

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