2002 AMC 10A Problem 2

The symbol $(a, b, c)$ is just a rule: first over second, plus second over third, plus third over first.

1. Match the numbers to the letters in order: $a = 2$, $b = 12$, $c = 9$.

2. Write the three fractions:
$$\frac{a}{b} = \frac{2}{12}, \qquad \frac{b}{c} = \frac{12}{9}, \qquad \frac{c}{a} = \frac{9}{2}.$$

3. Simplify each one first, because smaller numbers are easier to add:
$$\frac{2}{12} = \frac{1}{6}, \qquad \frac{12}{9} = \frac{4}{3}, \qquad \frac{9}{2} \text{ stays as it is}.$$

4. The denominators are $6$, $3$ and $2$, and all of them go into $6$. Rewrite each fraction in sixths:
$$\frac{1}{6} + \frac{8}{6} + \frac{27}{6} = \frac{36}{6} = 6.$$

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

A made-up symbol is a function in disguise: it tells you what to do with the inputs, in a fixed order. The only skills needed are careful substitution and adding fractions. Simplifying each fraction before finding a common denominator keeps the numbers small.

Estimate. $\frac{9}{2} = 4.5$, $\frac{12}{9}$ is a little more than $1$, and $\frac{2}{12}$ is a small piece. The total is a bit more than $5.5$ and well under $6.5$, and the only choice in that range is $6$.

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 $\boxed{\textbf{(C)}\ 6}$.