AMC 8 · 2022 · #2

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

function-evaluationorder-of-operationsformula-substitution identify-subproblems ↑ Prerequisites: order-of-operationsexponents

📏 Short solution 💡 1 insight

Problem

Two brand-new operations are defined: a ◆ b = a² - b² and a bigstar b = (a-b)². They behave like functions that take two numbers as inputs. The task is to evaluate (5 ◆ 3) bigstar 6, doing the inner operation first because of the parentheses.

Pick an answer.

(A)

${-}20$

(B)

(C)

(D)

100

(E)

220

AMC 8 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The expression has a clear inside-then-outside structure thanks to the parentheses, so Tool #7 (Identify Subproblems) is the natural fit: first compute the inner 5 ◆ 3, then plug that result into the outer bigstar 6. We treat each unfamiliar symbol like a tiny recipe — read the letters a and b, line them up with the actual numbers around the symbol, and substitute. Tool #3 (Eliminate Possibilities) is the AMC multiple-choice safety net: after computing, we check that the value matches one of the five offered choices and that the others can be ruled out, guarding against an arithmetic slip.

1STEP 1

Split at the parentheses into two subproblems: first find 5 ◆ 3, then apply bigstar 6 to that result.

(5 ◆ 3) bigstar 6 = [ 5 ◆ 3 ] bigstar 6

2STEP 2

Apply the ◆ rule with a = 5, b = 3: since a ◆ b = a² - b², substitute to write 5² - 3².

5 ◆ 3 = 5² - 3²

3STEP 3

Square and subtract: 5² - 3² = 25 - 9 = 16.

5² - 3² = 25 - 9 = 16

4STEP 4

Substitute back into a bigstar b = (a-b)²: 16 bigstar 6 = (16 - 6)² = 10² = 100.

16 bigstar 6 = (16 - 6)² = 10² = 100

5STEP 5

Among the choices -20, 4, 16, 100, 220 only 100 matches, so the answer is (D); 16 is the stop-halfway trap.

100 → (D)

Answer

100

The two operations both involve squaring, so the final answer should be a non-negative number on the order of 10². We got 100, which is exactly 10² and lands in the middle of the answer range — perfectly reasonable. We can also sanity-check the trap: choice (C) 16 is the inner result alone, choice (A) -20 would only appear if we accidentally did a - b² instead of (a-b)², and choice (E) 220 is roughly 16² - 6² = 220, the answer if someone mistakenly applied ◆ on the outside instead of bigstar. Each distractor maps to a specific misstep, and our path avoids all of them.

💡Key takeaway

This AMC 8 problem only needs Grade 6 expression evaluation — plug numbers into the letters of a formula — that you already know!

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