AMC 8 · 2008 · #8
Grade 6 arithmeticProblem
Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars?
Pick an answer.
AMC 8 2008 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.
Toolkit + CCSS Solution
Understand
Restated: A bar chart shows the Boosters Club's candy sales for January, February, March, and April. Find the average sales per month, in dollars.
Givens: Horizontal gridlines on the chart mark $\$40$, $\$80$, and $\$120$; January bar reaches halfway between $\$80$ and $\$120$; February bar reaches halfway between $\$40$ and $\$80$; March bar reaches $\$40$; April bar reaches $\$120$; Answer choices: (A) $60$, (B) $70$, (C) $75$, (D) $80$, (E) $85$
Unknowns: The average sales per month, in dollars
Understand
Restated: A bar chart shows the Boosters Club's candy sales for January, February, March, and April. Find the average sales per month, in dollars.
Givens: Horizontal gridlines on the chart mark $\$40$, $\$80$, and $\$120$; January bar reaches halfway between $\$80$ and $\$120$; February bar reaches halfway between $\$40$ and $\$80$; March bar reaches $\$40$; April bar reaches $\$120$; Answer choices: (A) $60$, (B) $70$, (C) $75$, (D) $80$, (E) $85$
Plan
Primary tool: #15 Visualize / Read the Diagram
Secondary: #2 Make a Systematic List
The numbers we need are not written out — they are encoded in the heights of the bars. Tool #15 (Visualize / Read the Diagram) is the first move: read each bar against the gridlines at $\$40$, $\$80$, and $\$120$ to pull out a number for each month. Tool #2 (Make a Systematic List) keeps the four month-by-month values lined up so we can add them with no mix-ups. After that the problem collapses to the Grade 6 mean formula: sum the four values and divide by $4$.
Execute — Answer: D
3.MD.B.3 Step 1 - Read each bar against the gridlines.
- The lines at $\$40$, $\$80$, and $\$120$ split the chart into equal $\$40$ chunks, so a bar halfway between two gridlines is $\$20$ above the lower one.
💡 Reading a scaled bar graph is the Grade 3 "draw and read bar graphs" skill. The halfway bars in Jan and Feb sit at $\$100$ and $\$60$.
4.NBT.B.4 Step 2 - List the four monthly sales and add them.
- Pairing $\$100 + \$120 = \$220$ and $\$60 + \$40 = \$100$ first keeps the arithmetic clean.
💡 A systematic list prevents skipping or double-counting a month. Adding in friendly pairs is the standard mental-math regrouping move.
6.SP.B.5 Step 3 - Apply the mean formula.
- There are $4$ months, so divide the total by $4$.
💡 Mean $=$ total $\div$ count. $320 \div 4 = 80$ because $4 \times 80 = 320$.
3.MD.B.3 Read each bar against the gridlines. The lines at $\$40$, $\$80$, and $\$120$ sp 4.NBT.B.4 List the four monthly sales and add them. Pairing $\$100 + \$120 = \$220$ and $\ 6.SP.B.5 Apply the mean formula. There are $4$ months, so divide the total by $4$. Review
Reasonableness: The four sales are $\$40$, $\$60$, $\$100$, $\$120$. The smallest is $\$40$ and the largest is $\$120$, so the average must lie strictly between $\$40$ and $\$120$. Our answer $\$80$ sits right in that range. It also lines up with eyeballing the chart: two bars are below $\$80$ (Mar and Feb) and two are above ($\$80$) (Jan and Apr), which is exactly what you expect when $\$80$ is the balance point.
Alternative: Tool #11 (Use Symmetry / Pair Off): pair Mar with Apr to get $40 + 120 = 160$, and pair Feb with Jan to get $60 + 100 = 160$. Each pair contributes the same total, so the four-month total is $2 \times 160 = \$320$ and the average is $\$320 \div 4 = \$80$. The pairing makes the division trivial and confirms (D).
CCSS standards used (min grade 6)
3.MD.B.3Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories; solve one- and two-step problems using information from the graphs (Reading each month's sales off the bar chart by comparing bar heights to the $\$40$ / $\$80$ / $\$120$ gridlines.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers using the standard algorithm (Adding the four monthly amounts $100 + 60 + 40 + 120 = 320$.)6.SP.B.5Summarize numerical data sets, including reporting the number of observations and measures of center (Applying the mean formula (sum $\div$ count) to get the average sales $\$320 \div 4 = \$80$.)
⭐ Read the bars, list the four numbers, add, and divide by $4$ — the mean is just "total split fairly" across the months.
⭐ Read the bars, list the four numbers, add, and divide by $4$ — the mean is just "total split fairly" across the months.
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