Competition · AMC preparation · step 4 of 4
AMC 8 · 2014 · #3
Grade 6 arithmeticPick an answer.
AMC 8 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The week splits naturally into three pieces — Days 1–3, Days 4–6, and Day 7 — with different reading paces, so Tool #7 (Identify Subproblems) is the cleanest entry: compute the page total for each piece, then add. Tool #2 (Find a Pattern) is the supporting move: noticing that the first two pieces share the same length (3 days) lets us factor and write 3 × (36 + 44) + 10 instead of doing two separate multiplications, which is faster and easier to check.
Turn the first average into pages
Average × days recovers a block total, so the first 3 days come to 108 pages.
The Grade 6 view of the mean: mean × count = total. Multiplying recovers the sum from the average.
Over the first three days, Isabella read 36 × 3 = 108 pages in all.
▸ Why?
An average of 36 pages a day for those 3 days means the block holds the same total as 3 days that each read exactly 36 pages, so its total is 36 counted three times.
▸ Why?
Replacing each day's real count with the average of 36 leaves the block's total unchanged, because the average is precisely the block's total pages shared equally over the 3 days — so the total is recovered as that equal share times the number of days (sum = mean × count).
▸ Why?
Three days that each read 36 pages form 3 equal groups of 36, and multiplying 36 by 3 is exactly how we count 3 equal groups.
Turn the second average into pages
The same move on the next 3 days: 44 pages/day gives 132 pages.
Same "mean times count" move applied to the second subproblem.
6.SP.A.3Identify SubproblemsAdd the three pieces
Add the three pieces — 108, 132, and the last day's 10 — for the whole book: 250 pages.
Combining the results of independent subproblems with addition is the Grade 4 multi-step word-problem skill.
4.OA.A.3Identify SubproblemsFactor out the shared 3
Cross-check: both blocks are 3 days, so factor once — 3 × (36 + 44) + 10 = 250.
Spotting the repeated factor (3 days, twice) turns two multiplications into one — the distributive property at work.
4.OA.A.3Make A Systematic ListThis AMC 8 problem only needs the Grade 6 idea that "average × count = total" — split the week into pieces, multiply each piece, then add.
- Turn the first average into pages
- Turn the second average into pages
- Add the three pieces
- Factor out the shared 3
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