AMC 10 · 2002 · #2

Grade 6 arithmetic

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

function-evaluationfraction-arithmetic formula-substitution ↑ Prerequisites: function-evaluation

📏 Short solution 💡 1 insight

Problem

A new operation is defined by

(a, b, c) = \frac{a}{b} + \frac{b}{c} + \frac{c}{a}

for non-zero real numbers

a

b

c

, and the order of the three inputs is fixed. Find the value of

(2, 12, 9)

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The strange-looking symbol (a, b, c) is just a recipe: it tells you to build three separate fractions and add them. Tool #7 (Identify Subproblems) handles this cleanly — substitute the numbers to get the three fractions, simplify each one on its own, then add them over a common denominator. Tool #3 (Eliminate Possibilities) is the safety net: since every answer choice is a whole number, a quick size estimate of the three pieces pins the total to a single choice and catches any slip in the fraction work.

1STEP 1

Substitute the numbers into the rule

Each letter names a slot, so with $a=2$ , $b=12$ , $c=9$ the recipe gives $(2, 12, 9)=\frac{2}{12}+\frac{12}{9}+\frac{9}{2}$ .

(2, 12, 9) = 2/12 + 12/9 + 9/2

2STEP 2

Simplify each fraction

Reduce first so the numbers stay small: $\frac{2}{12}=\frac{1}{6}$ and $\frac{12}{9}=\frac{4}{3}$ , while $\frac{9}{2}$ is already lowest.

2/12 = 1/6, 12/9 = 4/3, 9/2 = 9/2

3STEP 3

Add over a common denominator

The denominators 6, 3, 2 all fit into 6: $\frac{1}{6}+\frac{8}{6}+\frac{27}{6}=\frac{36}{6}$ , so the value is 6 — choice (C).

1/6 + 8/6 + 27/6 = 36/6 = 6 → (C)

Answer

A quick size estimate confirms the total. The three pieces are about 0.17, 1.33, and 4.5, and 0.17 + 1.33 + 4.5 ≈ 6. Because every answer choice is a whole number, the total must land exactly on 6 — the neighbours 5 and 7 are too far from the estimate. This also guards the order trap: if the fractions had been built in the wrong order the sum would not have come out to a clean integer near this estimate.

💡Key takeaway

A made-up operation is just a recipe: put each number where its letter goes, then do the ordinary fraction arithmetic carefully.

Substitute the numbers into the rule
Simplify each fraction
Add over a common denominator

Substitute the numbers into the rule

Simplify each fraction

Add over a common denominator

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