Competition · AMC preparation · step 4 of 4
AMC 8 · 2014 · #6
Grade 4 arithmeticPick an answer.
AMC 8 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Each rectangle's area is 2 × length, so the total area is 2 × 1 + 2 × 4 + … + 2 × 36. The shared factor 2 is a pattern (Tool #4) — every term has it — so we can factor it out and add the lengths just once instead of six times: total = 2 × (1 + 4 + 9 + 16 + 25 + 36). Tool #7 (Identify Subproblems) then splits the work into two clean pieces — first add the six numbers, then multiply by 2.
Write the area as six products
Write the total as a sum of six rectangle areas, each equal to width × length = 2 × its length.
Area = width × length for a rectangle is the Grade 3 area standard.
3.MD.C.7Identify SubproblemsPull out the common 2
Every term shares a factor 2 — pull it out by the distributive property, so the lengths get added just once.
Factoring out a common factor is exactly the distributive property of multiplication over addition.
Adding the six rectangle areas gives the same result as adding the six lengths first and then multiplying that single sum by the shared width 2.
▸ Why?
Each rectangle's area is its width 2 multiplied by its own length, so the total is 2·1 + 2·4 + 2·9 + 2·16 + 2·25 + 2·36 — every term is a 2 repeated as many times as that length.
▸ Why?
A rectangle 2 wide and ℓ long tiles into ℓ columns, each holding 2 unit squares, so its area is 2 counted ℓ times — a multiplication.
▸ Why?
Because every term in that sum carries the same factor 2, the 2 can be pulled out front: adding the lengths once and then multiplying by 2 builds the very same total, since a shared group can be split apart or joined back.
Add the six squares
Add the six squares by easy pairs: (1 + 9) + (4 + 16) + (25 + 36) = 10 + 20 + 61 = 91.
Pairing numbers that add to round values (like 10 and 20) is a standard mental-math regrouping move.
4.NBT.B.4Identify SubproblemsMultiply by the width
Multiply the sum by the common width: 2 × 91 = 182 → (D).
A one-digit by two-digit multiplication is Grade 4 multi-digit arithmetic.
4.NBT.B.5Identify SubproblemsThis AMC 8 problem only needs Grade 4 arithmetic — once you factor out the shared width, it's just one addition and one multiplication!
- Write the area as six products
- Pull out the common 2
- Add the six squares
- Multiply by the width
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