2002 AMC 10A Problem 2

Match the letters to the numbers in order: $a = 2$, $b = 12$, $c = 9$. Then
$$(2, 12, 9) = \frac{2}{12} + \frac{12}{9} + \frac{9}{2} = \frac{1}{6} + \frac{4}{3} + \frac{9}{2}.$$

The least common denominator is $6$:
$$\frac{1}{6} + \frac{8}{6} + \frac{27}{6} = \frac{36}{6} = 6.$$

The answer is $\boxed{\textbf{(C)}\ 6}$.

A newly defined symbol is just a function of its inputs; the only task is careful substitution in the stated order, followed by fraction arithmetic. Reducing each fraction first ($\tfrac{2}{12} = \tfrac16$, $\tfrac{12}{9} = \tfrac43$) keeps the common denominator small.

Estimate: $\tfrac{9}{2} = 4.5$, $\tfrac{12}{9}$ is a bit more than $1$, and $\tfrac{2}{12}$ is small. The total is a little above $5.5$, and the only integer choice that fits is $6$.

• Mixing up the order of the arguments (for example computing 2/9 or 12/2) so the fractions do not match the definition.
• Arithmetic slips when converting to the common denominator, such as writing $\tfrac{9}{2} = \tfrac{18}{6}$.