Competition · AMC preparation · step 4 of 4

AMC 10 · 2007A · #2

Grade 7 arithmetic

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

function-evaluationformula-substitutionorder-of-operations formula-substitution ↑ Prerequisites: order-of-operations

📏 Short solution 💡 1 insight

Problem

Two made-up operations are defined by rules: a@b = ab - b² and a#b = a + b - ab². Using these rules, find the value of the fraction 6@2/6#2.

Pick an answer.

(A)

$- \frac {1}{2}$

(B)

$- \frac {1}{4}$

(C)

$\frac {1}{8}$

(D)

$\frac {1}{4}$

(E)

$\frac {1}{2}$

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

A fraction is just a top divided by a bottom, so Tool #7 (Identify Subproblems) splits the job into three clean pieces: compute the numerator 6@2, compute the denominator 6#2, then divide. Each custom operation is only a substitution into its rule. Tool #3 (Eliminate Possibilities) gives a quick sanity check: once the top comes out positive and the bottom negative, the answer must be negative, which already rules out three of the five choices.

1STEP 1

Read the two rules

Each symbol is a recipe: a marks the first number, b the second, so both 6@2 and 6#2 take a = 6 and b = 2.

a = 6, b = 2

2STEP 2

Compute the numerator

Apply @: the power first, 2² = 4, then 6 × 2 = 12, so the numerator is 12 - 4 = 8.

6@2 = 6 · 2 - 2² = 12 - 4 = 8

3STEP 3

Compute the denominator

Apply #: 6·2² = 6 × 4 = 24 while 6 + 2 = 8, so the denominator is 8 - 24 = -16.

6#2 = 6 + 2 - 6 · 2² = 8 - 24 = -16

4STEP 4

Divide top by bottom

Now 6@2/6#2 = 8/-16, and a positive over a negative of twice the size gives -1/2 — choice (A).

6@2/6#2 = 8/-16 = -1/2 → (A)

Answer

- 1/2

Recheck each piece: 6@2 = 12 - 4 = 8 and 6#2 = 8 - 24 = -16, so the ratio is 8/(-16) = -1/2. The numerator is positive and the denominator is negative, so the answer has to be negative — that alone leaves only (A) -1/2 and (B) -1/4, and the exact division picks (A). The size is also sensible: 8 is half of 16, so the magnitude 1/2 is expected.

💡Key takeaway

A strange new symbol is just a recipe: put the numbers where the letters are, follow order of operations, and work the top and bottom of a fraction one at a time.

Read the two rules
Compute the numerator
Compute the denominator
Divide top by bottom

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Read the two rules

Compute the numerator

Compute the denominator

Divide top by bottom

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