AMC 10 · 2004 · #2

Grade 7 arithmetic

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

function-evaluationfraction-arithmetic identify-subproblems ↑ Prerequisites: order-of-operations

📏 Medium solution 💡 1 insight

Problem

A three-input operation is defined by ⊗(a,b,c)=a/(b-c), valid whenever b ≠ c. Evaluate the nested expression ⊗(⊗(1,2,3), ⊗(2,3,1), ⊗(3,1,2)).

Pick an answer.

(A)

$-\frac{1}{2}$

(B)

$-\frac{1}{4}$

(C)

(D)

$\frac{1}{4}$

(E)

$\frac{1}{2}$

AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The expression is one operation wrapped around three copies of itself, so it splits into clean subproblems (Tool #7): first evaluate the three inner ⊗ calls, then feed those three numbers into the outer ⊗. The whole trap is signs — two inner calls come out negative, and the outer bottom is a subtraction of a negative. Tool #3 (Eliminate Possibilities) closes it: once the result is a negative fraction, the positive choices 0, 1/4, 1/2 are gone, and only its size decides between -1/2 and -1/4.

1STEP 1

Decode the operation

The rule is a machine: first number on top, second minus third on the bottom. Inner calls come first, like parentheses inside parentheses.

⊗(a,b,c)=a/(b-c)

2STEP 2

Evaluate the three inner calls

Apply the rule to each inner triple: ⊗(1,2,3)=-1, ⊗(2,3,1)=1, ⊗(3,1,2)=-3. Two bottoms are negative, so two results are.

⊗(1,2,3)=-1, ⊗(2,3,1)=1, ⊗(3,1,2)=-3

3STEP 3

Set up the outer call

Feed -1, 1, -3 in order into ⊗(-1,1,-3)=-1/(1-(-3)). Subtracting -3 adds 3, so the bottom is 4.

⊗(-1,1,-3)=-1/(1-(-3))=-1/4

4STEP 4

Finish and rule out the traps

The outer value is -1/4 — negative over positive. Sign kills 0, 1/4, 1/2; -1/2 would need a bottom of 2. So (B).

-1/4=-1/4 → (B)

Answer

-1/4

The final fraction is -1/4: a small negative number near -0.25. It should be negative because the top (-1) is negative and the bottom (4) is positive, which immediately rejects the non-negative choices. Its size is a quarter, not a half, so -1/2 is too big and -1/4 fits. Both checks land on (B).

💡Key takeaway

When an operation is nested inside itself, finish every inner piece first, then drop those numbers into the outer rule — and watch the minus signs at each step.

Decode the operation
Evaluate the three inner calls
Set up the outer call
Finish and rule out the traps

Decode the operation

Evaluate the three inner calls

Set up the outer call

Finish and rule out the traps

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