AMC 8 · 2013 · #6

Grade 4 arithmeticalgebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

linear-equations-one-varfactorsmental-arithmetic convert-to-algebraidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticfactors

📏 Medium solution 💡 2 insights 📊 Diagram

📘 View easy version →

Problem

The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, $30 = 6\times5$ . What is the missing number in the top row?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2013 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.

View mode:

Toolkit + CCSS Solution

Understand

Restated: A three-row pyramid of boxes has the rule: every box equals the product of the two boxes directly above it that touch it. The top row reads $6, 5, ?$; the middle row reads $30, ?$; the bottom row reads $600$. Find the missing number in the top row.

Givens: Top row left-to-right: $6, 5, y$ (where $y$ is the unknown to find); Middle row left-to-right: $30, x$ (where $x$ is the other unknown middle box); Bottom row: $600$; Pyramid rule: each box $= $ product of the two boxes touching it from above; Given confirmation: $30 = 6 \times 5$; Answer choices: (A) $2$, (B) $3$, (C) $4$, (D) $5$, (E) $6$

Unknowns: The missing top-row number $y$ (the rightmost top box)

Understand

Plan

Primary tool: #11 Work Backwards

Secondary: #7 Identify Subproblems

The rule moves information downward (top numbers $\to$ middle $\to$ bottom by multiplication). We know the bottom value $600$ and one middle value $30$, so the most direct route is Tool #11 (Work Backwards): reverse each multiplication into a division as we climb from the bottom to the top. Tool #7 (Identify Subproblems) splits the climb into two independent one-step subproblems: first $600 \div 30$ to recover the missing middle box, then that result $\div 5$ to recover the missing top box. No algebraic variables are needed — just two divisions in the right order.

Execute — Answer: C

#11 Work Backwards 3.OA.B.6 Step 1

Subproblem 1 — climb from the bottom row to the middle row.
The bottom box $600$ is the product of the two middle boxes $30$ and $x$ (the unknown middle box).
Working backwards, reverse the multiplication: $x = 600 \div 30$.

$$30 \times x = 600 \;\Longrightarrow\; x = 600 \div 30 = 20$$

💡 Asking "$30$ times what gives $600$?" is exactly the Grade 3 unknown-factor view of division — division as the inverse of multiplication.

#7 Identify Subproblems 3.OA.B.6 Step 2

Subproblem 2 — climb from the middle row to the top row.
The middle box we just found ($x = 20$) is the product of the two top boxes that touch it: $5$ (known) and $y$ (the answer we want).
Reverse the multiplication again: $y = 20 \div 5$.

$$5 \times y = 20 \;\Longrightarrow\; y = 20 \div 5 = 4$$

💡 Splitting the pyramid into two single-step "reverse the product" pieces is the Tool #7 subproblem move — and each piece is again Grade 3 unknown-factor division.

#11 Work Backwards 4.OA.A.3 Step 3

Read off the answer.
The missing top-row number is $y = 4$, which matches choice (C).

$$y = 4 \;\Rightarrow\; \textbf{(C)}$$

💡 Chaining two reverse-multiplication steps to solve a multi-step word problem is the Grade 4 expectation for word-problem reasoning.

[1] #11 3.OA.B.6 Subproblem 1 — climb from the bottom row to the middle row. The bottom box $600$

[2] #7 3.OA.B.6 Subproblem 2 — climb from the middle row to the top row. The middle box we just

[3] #11 4.OA.A.3 Read off the answer. The missing top-row number is $y = 4$, which matches choice

Review

Reasonableness: Rebuild the pyramid top-down with our answers and check every box. Top row: $6, 5, 4$. Middle row: $6 \times 5 = 30$ (matches given) and $5 \times 4 = 20$. Bottom row: $30 \times 20 = 600$ (matches given). Every rule is satisfied, so $y = 4$ is consistent.

Alternative: Tool #6 (Guess and Check) on the five choices. For each candidate $y$, compute the right middle box $5 \times y$ and then the bottom box $30 \times (5 \times y) = 150 y$. We need $150 y = 600$, i.e. $y = 4$. Quick scan: $y=2 \to 300$, $y=3 \to 450$, $y=4 \to 600$ ✓, $y=5 \to 750$, $y=6 \to 900$. Only (C) works.

CCSS standards used (min grade 4)

3.OA.B.6 Understand division as an unknown-factor problem (Reversing each multiplication step: "$30 \times \;? = 600$" gives $? = 600 \div 30 = 20$, and "$5 \times \;? = 20$" gives $? = 20 \div 5 = 4$.)
4.OA.A.3 Solve multi-step word problems using the four operations (Chaining the two reverse-multiplication subproblems in the correct order (bottom $\to$ middle, then middle $\to$ top) to reach the final answer.)

⭐ Climb the pyramid backwards: if multiplication built it going down, division un-builds it going up — and that is just Grade 3 "$30$ times what equals $600$?" thinking.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 4)

More like this