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

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Problem

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.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Work Backwards

The rule moves information downward (top numbers → middle → 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 ÷ 30 to recover the missing middle box, then that result ÷ 5 to recover the missing top box. No algebraic variables are needed — just two divisions in the right order.

1STEP 1

The bottom box 600 is 30 times the empty middle box, so reverse it: 600 ÷ 30 = 20.

30 × x = 600 ⟹ x = 600 ÷ 30 = 20

2STEP 2

That middle 20 is 5 times the top-right box it touches, so reverse again: 20 ÷ 5 = 4.

5 × y = 20 ⟹ y = 20 ÷ 5 = 4

3STEP 3

The missing top-row number is 4 — choice (C).

y = 4 → (C)

Answer

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

💡Key takeaway

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.

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