2002 AMC 10A Problem 2

Problem 2 of 25EasierAlgebra

For the nonzero numbers

a,

b,

and

c,

define

(a,b,c)=\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}.

Find

(2,12,9).

Show solution

Solution

\begin{aligned} (2,12,9) &= \dfrac{2}{12}+\dfrac{12}{9}+\dfrac{9}{2} \\ &= \dfrac{1}{6}+\dfrac{4}{3}+\dfrac{9}{2}. \end{aligned}

Over a denominator of

6,

this is

\dfrac{1+8+27}{6}=\dfrac{36}{6}=6.

Thus, the correct answer is C.

Concepts: custom operation · fraction

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.