AMC 10 · 2005 · #2

Grade 7 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

function-evaluationfraction-arithmeticorder-of-operations identify-subproblemsformula-substitution ↑ Prerequisites: fraction-arithmeticorder-of-operations

📏 Short solution 💡 1 insight

Problem

A new operation ★ is defined for real numbers a ≠ b by (a★ b)=(a+b)/(a-b). Working from the inside out, evaluate ((1★ 2)★ 3), and decide whether the result is defined.

Pick an answer.

(A)

$-\frac{2}{3}$

(B)

$-\frac{1}{5}$

(C)

(D)

$\frac{1}{2}$

(E)

This value is not defined.

AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The expression is one operation nested inside another, so Tool #7 (Identify Subproblems) splits it into two clean pieces: first compute the inner value 1★ 2, then feed that result into (·)★ 3. Tool #3 (Eliminate Possibilities) handles the only trap: choice (E) says the value is not defined, which happens only if a denominator a-b equals 0; checking each denominator confirms whether (E) can be dismissed.

1STEP 1

Plug into the inner operation

Inside first: a=1, b=2 makes the top 1+2=3 and the bottom 1-2=-1, so 1★ 2=3/-1 — legal because 1 ≠ 2.

1★ 2=(1+2)/(1-2)=3/-1

2STEP 2

Finish the inner value

A positive over a negative is negative, so 3/-1=-3, and that value becomes the left input of the outer ★.

3/-1=-3 → (1★ 2)=-3

3STEP 3

Evaluate the outer operation

Now (-3)★ 3: the top -3+3 vanishes while the bottom -3-3=-6 does not, so the value is defined and equals 0, choice (C).

(-3)★ 3=(-3+3)/(-3-3)=0/-6=0 → (C)

Answer

Both denominators were checked to be nonzero (1-2=-1 and -3-3=-6), so the operation was legal at every step and choice (E) is correctly eliminated. The final answer hinges on the outer numerator -3+3 being 0; adding a number to its opposite always gives 0, which is a reliable way to land on exactly 0. Among the choices only (C) is 0, and the other numeric choices would each require a nonzero numerator, which the computation does not produce.

💡Key takeaway

When an operation is spelled out as a formula, plug the numbers in innermost first, and remember a fraction is 0 exactly when its top is 0 and its bottom is not.

Plug into the inner operation
Finish the inner value
Evaluate the outer operation

Plug into the inner operation

Finish the inner value

Evaluate the outer operation

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