Competition · AMC preparation · step 4 of 4
AMC 8 · 2013 · #6
Grade 4 arithmeticalgebra
Pick an answer.
AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Climb to the middle row
The bottom box 600 is 30 times the empty middle box, so reverse it: 600 ÷ 30 = 20.
Asking "30 times what gives 600?" is exactly the Grade 3 unknown-factor view of division — division as the inverse of multiplication.
The unknown middle-row box equals 600 ÷ 30.
▸ Why?
The bottom box 600 is the product of the two middle boxes, so 30 times the unknown middle box equals 600; that makes the unknown box the missing factor, and a missing factor is recovered by dividing the product by the factor you already know.
▸ Why?
Division is exactly what reverses multiplication, so undoing "times 30" is done by " ÷ 30", turning the product 600 back into the number that was multiplied by 30.
Climb to the top row
That middle 20 is 5 times the top-right box it touches, so reverse again: 20 ÷ 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.
3.OA.B.6Identify SubproblemsRead off the answer
The missing top-row number is 4 — choice (C).
Chaining two reverse-multiplication steps to solve a multi-step word problem is the Grade 4 expectation for word-problem reasoning.
4.OA.A.3Work BackwardsClimb 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.
- Climb to the middle row
- Climb to the top row
- Read off the answer
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