AMC 8 · 2017 · #1

Grade 5 arithmetic
Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In
order-of-operationsmental-arithmetic systematic-enumeration ↑ Prerequisites: multi-digit-arithmetic
📏 Short solution 💡 2 insights
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Problem
Five short arithmetic expressions are listed as answer choices (A)-(E). Each one uses the digits 2, 0, 1, 7 glued together with + and × signs in different spots. The job is to evaluate every expression using the standard order of operations and pick the one with the largest value.

Pick an answer.

(A)
2+0+1+7
(B)
$2 \times 0 +1+7$
(C)
$2+0 \times 1 + 7$
(D)
$2+0+1 \times 7$
(E)
$2 \times 0 \times 1 \times 7$

AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Eliminate Possibilities

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.

1STEP 1

(A) 2+0+1+7 has only addition, so add the four digits to get 10.

2+0+1+7 = 10
2STEP 2

(B) 2 × 0 + 1 + 7: do the × first (2 × 0 = 0), then add to get 8.

2 × 0 + 1 + 7 = 0 + 1 + 7 = 8
3STEP 3

(C) 2 + 0 × 1 + 7: multiply first (0 × 1 = 0), then add to get 9.

2 + 0 × 1 + 7 = 2 + 0 + 7 = 9
4STEP 4

(D) 2 + 0 + 1 × 7: multiply first (1 × 7 = 7), then add to get 9 — not the 21 you get by ignoring order.

2 + 0 + 1 × 7 = 2 + 0 + 7 = 9
5STEP 5

(E) 2 × 0 × 1 × 7 is all multiplication, and any product containing a 0 is 0.

2 × 0 × 1 × 7 = 0
6STEP 6

Compare 10, 8, 9, 9, 0 — the largest is 10, so (A) wins.

max(10, 8, 9, 9, 0) = 10 → (A)
Answer
2+0+1+7
Intuitively, (A) keeps every digit alive by only adding, while every other choice either multiplies by 0 (which zeroes out a term) or replaces an addition with a multiplication-by-1 (which doesn't help). So pure addition should be the biggest, and indeed 10 beats 8, 9, 9, 0. The answer (A) matches that intuition.
💡Key takeaway

This AMC 8 problem only needs Grade 5 order of operations (multiplication before addition) that you already know!

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