AMC 8 · 2014 · #3
Easy mode Grade 6Problem
Isabella read a book over days.
On each of the first days, she read about pages.
On each of the next days, she read about pages.
On the th day, she read pages and finished the book.
How many pages were in the whole book?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: Isabella read a book over $7$ days. For the first $3$ days she averaged $36$ pages per day, for the next $3$ days she averaged $44$ pages per day, and on the last day she finished the book by reading $10$ pages. How many pages were in the book in total?
Givens: Days $1$–$3$: average of $36$ pages per day; Days $4$–$6$: average of $44$ pages per day; Day $7$: $10$ pages read (finishes the book); Answer choices: (A) $240$, (B) $250$, (C) $260$, (D) $270$, (E) $280$
Unknowns: The total number of pages in the book
Understand
Restated: Isabella read a book over $7$ days. For the first $3$ days she averaged $36$ pages per day, for the next $3$ days she averaged $44$ pages per day, and on the last day she finished the book by reading $10$ pages. How many pages were in the book in total?
Givens: Days $1$–$3$: average of $36$ pages per day; Days $4$–$6$: average of $44$ pages per day; Day $7$: $10$ pages read (finishes the book); Answer choices: (A) $240$, (B) $250$, (C) $260$, (D) $270$, (E) $280$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #2 Find a Pattern
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 $3 \times (36 + 44) + 10$ instead of doing two separate multiplications, which is faster and easier to check.
Execute — Answer: B
6.SP.A.3 Step 1 - Convert the first average into a page total.
- An average of $36$ pages per day for $3$ days means $36 \times 3$ pages were read in that block.
💡 The Grade 6 view of the mean: $\text{mean} \times \text{count} = \text{total}$. Multiplying recovers the sum from the average.
6.SP.A.3 Step 2 - Do the same for the second block.
- An average of $44$ pages per day for $3$ days gives $44 \times 3$ pages.
💡 Same "mean times count" move applied to the second subproblem.
4.OA.A.3 Step 3 Add the three pieces — the two $3$-day blocks and the single last day — to get the whole book.
💡 Combining the results of independent subproblems with addition is the Grade 4 multi-step word-problem skill.
4.OA.A.3 Step 4 Cross-check with the pattern trick: the first two blocks share the same $3$-day length, so factor $3$ out and only multiply once.
💡 Spotting the repeated factor ($3$ days, twice) turns two multiplications into one — the distributive property at work.
6.SP.A.3 Convert the first average into a page total. An average of $36$ pages per day fo 6.SP.A.3 Do the same for the second block. An average of $44$ pages per day for $3$ days 4.OA.A.3 Add the three pieces — the two $3$-day blocks and the single last day — to get t 4.OA.A.3 Cross-check with the pattern trick: the first two blocks share the same $3$-day Review
Reasonableness: Across the full $7$ days she reads $250$ pages, which is an overall average of $250 / 7 \approx 35.7$ pages per day. That sits sensibly between the $36$ pages/day pace and the $44$ pages/day pace — pulled down by the small $10$-page final day. A typical school book of around $250$ pages is also realistic for a one-week assignment.
Alternative: Tool #6 (Guess and Check) on the answer choices: from the structure, the total must equal $3 \times 36 + 3 \times 44 + 10$, which is a fixed sum, not a range — so plugging the choices isn't needed. But as a sanity test, $(A)\,240$ would force the last day to be $0$ pages, $(C)\,260$ would require $20$ on the last day, $(D)\,270$ would require $30$, and $(E)\,280$ would require $40$. Only $(B)\,250$ matches the stated $10$ pages on Day $7$.
CCSS standards used (min grade 6)
3.OA.A.1Interpret products of whole numbers (Reading "$36$ pages per day for $3$ days" as the product $36 \times 3$, and likewise $44 \times 3$.)4.OA.A.3Solve multistep word problems with the four operations (Combining the three subtotals ($108 + 132 + 10$) and using the distributive shortcut $3 \times (36 + 44) + 10$ to reach the final total.)6.SP.A.3Recognize that a measure of center summarizes a data set with a single number (Using the relationship $\text{mean} \times \text{count} = \text{total}$ to turn each daily average into a page total for the block.)
⭐ This AMC 8 problem only needs the Grade 6 idea that "average $\times$ count = total" — split the week into pieces, multiply each piece, then add.
⭐ This AMC 8 problem only needs the Grade 6 idea that "average $\times$ count = total" — split the week into pieces, multiply each piece, then add.