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 rule named D takes three nonzero numbers and returns D(a,b,c)=abc/(a+b+c) — the product of the three numbers divided by their sum. Compute the value of D(2,4,6).

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 formula is one fraction built from two independent pieces — a product on top and a sum on the bottom — so Tool #7 (Identify Subproblems) says: compute the numerator and the denominator on their own, then divide. Tool #3 (Eliminate Possibilities) flags the built-in trap: 24 is exactly the top-only product 2·4·6, so a solver who forgets to divide lands on (E); keeping the denominator in view rules that out.

1STEP 1

Substitute the inputs into the rule

Drop each given number into its letter's slot: D(a,b,c)=abc/(a+b+c) becomes (2·4·6)/(2+4+6).

D(2,4,6)=(2·4·6)/(2+4+6)

2STEP 2

Compute the numerator (product)

The top is the product 2·4·6, so multiply two at a time: 2·4=8, then 8·6=48.

2·4·6 = 8·6 = 48

3STEP 3

Compute the denominator (sum)

The bottom is the sum 2+4+6, so add two at a time: 2+4=6, then 6+6=12.

2+4+6 = 6+6 = 12

4STEP 4

Divide top by bottom

Now divide the top by the bottom: 48÷12=4 because 12·4=48, which is choice (C).

48/12=4 → (C)

Answer

A quick sanity check: the numerator 48 is a few times bigger than the denominator 12, so the answer should be a small whole number, not a huge one — 4 fits, while 24 (choice E) would need the denominator to be 2, which it is not. Also 48÷12=4 reverses cleanly to 12·4=48, confirming the division. The three tempting wrong answers each drop a piece of the work: 24 skips dividing, 6 is just the sum-adjacent trap, and 1 or 2 come from mis-adding or mis-multiplying.

💡Key takeaway

To use a formula, drop each number into its letter's spot, then finish the top and the bottom separately before you divide.

Substitute the inputs into the rule
Compute the numerator (product)
Compute the denominator (sum)
Divide top by bottom

Substitute the inputs into the rule

Compute the numerator (product)

Compute the denominator (sum)

Divide top by bottom

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