Competition · AMC preparation · step 4 of 4
AMC 8 · 2017 · #1
Grade 5 arithmeticPick an answer.
AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The five answer choices ARE the five candidates to test, which is the textbook setup for Tool #3 (Eliminate Possibilities) — compute each candidate's value, then keep only the winner. Tool #7 (Identify Subproblems) is the natural helper: each expression is its own self-contained mini-arithmetic problem, so we solve five tiny subproblems and compare. Algebra (Tool #13) is overkill here; the work is pure order-of-operations bookkeeping.
Evaluate choice A
(A) 2+0+1+7 has only addition, so add the four digits to get 10.
Adding four small whole numbers within 20 is a Grade 2 fluency skill.
2.NBT.B.5Identify SubproblemsEvaluate choice B
(B) 2 × 0 + 1 + 7: do the × first (2 × 0 = 0), then add to get 8.
Knowing to do × before + in a mixed expression is the Grade 5 order-of-operations idea.
5.OA.A.1Identify SubproblemsEvaluate choice C
(C) 2 + 0 × 1 + 7: multiply first (0 × 1 = 0), then add to get 9.
The hidden 0 × 1 in the middle only "counts" after order of operations is applied.
5.OA.A.1Identify SubproblemsEvaluate choice D
(D) 2 + 0 + 1 × 7: multiply first (1 × 7 = 7), then add to get 9 — not the 21 you get by ignoring order.
Order of operations protects you from the 21 trap and gives the correct value 9.
5.OA.A.1Identify SubproblemsEvaluate choice E
(E) 2 × 0 × 1 × 7 is all multiplication, and any product containing a 0 is 0.
The zero-product property — "any number times 0 is 0" — is a Grade 3 property of multiplication.
3.OA.B.5Identify SubproblemsPick the largest value
Compare 10, 8, 9, 9, 0 — the largest is 10, so (A) wins.
Comparing five whole numbers within 20 to pick the largest is a Grade 4 multi-digit comparison skill.
Among the five listed expressions, the one that joins the digits 2, 0, 1, 7 with only plus signs has the largest value.
▸ Why?
The plus-only expression reaches the full running total of all four digits, since adding keeps each digit's whole value in the sum.
▸ Why?
When every digit is only added, each one hands over its entire amount and nothing is discarded, so the parts rebuild the complete total.
▸ Why?
Each of the other four expressions must resolve a multiplication first, and that product never lands higher than just adding those digits would, so none can overtake the plus-only total.
▸ Why?
The rules of arithmetic fix that a multiplication is carried out before the additions around it, so in each other expression the marked product is settled first and sets the value it contributes.
▸ Why?
A product that contains the digit 0 collapses to 0, wiping out the value an addition of that digit would have kept, which drops the two times-zero expressions below the plus-only total.
▸ Why?
Multiplying builds equal groups, and groups of 0 — or zero groups of any number — hold nothing at all.
▸ Why?
A product that multiplies a digit by 1 just gives that digit back, so trading an addition for a times-one adds nothing extra and the total cannot rise above the plus-only sum.
▸ Why?
One group of a number is exactly that number, so multiplying by 1 leaves it unchanged.
This AMC 8 problem only needs Grade 5 order of operations (multiplication before addition) that you already know!
- Evaluate choice A
- Evaluate choice B
- Evaluate choice C
- Evaluate choice D
- Evaluate choice E
- Pick the largest value
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