AMC 10 · 2022 · #1

Grade 6 arithmetic

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

absolute-valueorder-of-operationsfunction-evaluation identify-subproblems ↑ Prerequisites: order-of-operationsabsolute-value

📏 Short solution 💡 1 insight

Problem

A new symbol is defined: x ◆ y = |x - y|, the distance between two numbers. We must compute (1 ◆ (2 ◆ 3)) - ((1 ◆ 2) ◆ 3), where the parentheses tell us which pair to combine first on each side.

Pick an answer.

(A)

${-}2$

(B)

${-}1$

(C)

(D)

(E)

AMC 10 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The expression has two separate halves joined by a minus sign, and each half has an inner ◆ that must be done first. Tool #7 (Subproblems) says: compute the left half on its own, compute the right half on its own, then subtract. The order-of-operations rule plus the absolute-value definition do all the work — no algebra needed. Tool #3 (Eliminate) is the multiple-choice safety net: once we have a numerical value, we read off the matching letter.

1STEP 1

Left side, inner pair first: 2 ◆ 3 = |2 - 3| = 1.

2 ◆ 3 = |2 - 3| = 1

2STEP 2

Left side done: 1 ◆ (2 ◆ 3) = 1 ◆ 1 = |1 - 1| = 0.

1 ◆ 1 = |1 - 1| = 0

3STEP 3

Right side, inner pair first: 1 ◆ 2 = |1 - 2| = 1.

1 ◆ 2 = |1 - 2| = 1

4STEP 4

Right side done: (1 ◆ 2) ◆ 3 = 1 ◆ 3 = |1 - 3| = 2.

1 ◆ 3 = |1 - 3| = 2

5STEP 5

Subtract the halves: 0 - 2 = -2, which matches choice (A).

0 - 2 = -2 → (A)

Answer

-2

Sanity check the two halves. Left half closes with 1 ◆ 1, and any number ◆ itself is 0 — that's the smallest possible value. Right half closes with 1 ◆ 3, two numbers that sit 2 apart, so the value is 2. A small number minus a bigger number gives a negative result, so 0 - 2 = -2 has the right sign and size to match choice (A).

💡Key takeaway

This AMC 10 problem only needs Grade 6 "absolute value is just the distance between two numbers" — once you read the parentheses carefully, the two halves come out to 0 and 2, and the difference is -2.

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