AMC 8 · 2014 · #3

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

mean-median-mode-rangemulti-digit-arithmetic identify-subproblems ↑ Prerequisites: multi-digit-arithmeticmean-median-mode-range

📏 Short solution 💡 2 insights

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Problem

Isabella had a week to read a book for a school assignment. She read an average of $36$ pages per day for the first three days and an average of $44$ pages per day for the next three days. She then finished the book by reading $10$ pages on the last day. How many pages were in the book?

Pick an answer.

(A)

240

(B)

250

(C)

260

(D)

270

(E)

280

AMC 8 2014 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.

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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

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

#7 Identify Subproblems 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.

$$36 \times 3 = 108 \text{ pages}$$

💡 The Grade 6 view of the mean: $\text{mean} \times \text{count} = \text{total}$. Multiplying recovers the sum from the average.

#7 Identify Subproblems 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.

$$44 \times 3 = 132 \text{ pages}$$

💡 Same "mean times count" move applied to the second subproblem.

#7 Identify Subproblems 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.

$$108 + 132 + 10 = 250 \text{ pages} \;\Rightarrow\; \textbf{(B)}$$

💡 Combining the results of independent subproblems with addition is the Grade 4 multi-step word-problem skill.

#2 Find a Pattern 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.

$$3 \times (36 + 44) + 10 = 3 \times 80 + 10 = 240 + 10 = 250$$

💡 Spotting the repeated factor ($3$ days, twice) turns two multiplications into one — the distributive property at work.

[1] #7 6.SP.A.3 Convert the first average into a page total. An average of $36$ pages per day fo

[2] #7 6.SP.A.3 Do the same for the second block. An average of $44$ pages per day for $3$ days

[3] #7 4.OA.A.3 Add the three pieces — the two $3$-day blocks and the single last day — to get t

[4] #2 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.1 Interpret 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.3 Solve 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.3 Recognize 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.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 6)

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