Competition · AMC preparation · step 4 of 4
AMC 8 · 2011 · #11
Grade 6 arithmetic
Pick an answer.
AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The question "how many more minutes per day on average" is really two smaller jobs glued together: (1) for each of the 5 days, find Sasha-minus-Asha; (2) average those 5 numbers. Tool #7 (Identify Subproblems) makes that split explicit so we don't try to eyeball the whole graph at once. Tool #16 (Change Focus) gives a clean cross-check: instead of averaging differences, we can total each girl's week first and take the difference of totals divided by 5 — these two views must agree, which catches arithmetic slips.
Find each day's difference
Read each day off the graph and compute Sasha minus Asha; days when Sasha studied less just count as negative.
Reading a scaled bar graph one category at a time to answer a comparison question is exactly the Grade 3 bar-graph standard.
3.MD.B.3Identify SubproblemsAdd the five differences
Add the five differences; the positive days give 60 and the negative days give -30, for a total of 30.
Adding multi-digit whole numbers (and their opposites) fluently is the Grade 4 NBT skill.
4.NBT.B.4Identify SubproblemsDivide by the number of days
Divide that total by the 5 days; the quotient is the average extra minutes per day.
The mean of a small data set = sum ÷ count is the Grade 6 "measure of center" definition.
The average extra minutes per day is the week's total extra minutes shared equally across the days, so it equals that total divided by the number of days.
▸ Why?
The average is the equal daily share — the one amount that, if it appeared on every one of the five days, would rebuild the same weekly total of extra minutes.
▸ Why?
If one equal share repeated over the five days rebuilds the total, then five equal shares of that size make the total, so the share is the total split into five equal groups.
▸ Why?
Splitting a total into a set number of equal groups is division, which reverses building that total up as that many equal groups multiplied together.
▸ Why?
The weekly total extra is just the five daily extras collected with none left out and none counted twice, so those parts add back to that one total.
This AMC 8 problem only needs Grade 6 "mean = sum ÷ count" — averaging is just adding and dividing!
- Find each day's difference
- Add the five differences
- Divide by the number of days
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