2005 AMC 10A Problem 2

The new operation is just a rule: add the two numbers on top, subtract them on the bottom. Parentheses tell you which part to do first, so start inside.

1. Inner part. Put $a = 1$ and $b = 2$ into the rule:
$$1 \star 2 = \frac{1 + 2}{1 - 2} = \frac{3}{-1} = -3.$$
2. Outer part. Now the left number is $-3$ and the right number is $3$:
$$(-3) \star 3 = \frac{-3 + 3}{-3 - 3} = \frac{0}{-6} = 0.$$
3. Is anything undefined? A fraction breaks only when its bottom is $0$. The bottoms were $-1$ and $-6$, so both steps are fine. A top of $0$ is allowed; it just makes the fraction equal $0$.

The answer is $\boxed{\textbf{(C)}\ 0}$.

A made-up symbol like $\star$ is a recipe: plug in the numbers and follow the steps, innermost parentheses first. When a choice says "not defined," ask whether any denominator is actually zero. Here the rule only uses pairs with $a \neq b$, so the bottom $a - b$ is never zero.

Work from the inside out. First,
$$1 \star 2 = \frac{1 + 2}{1 - 2} = \frac{3}{-1} = -3.$$
Next, substitute this result as the left input:
$$(-3) \star 3 = \frac{-3 + 3}{-3 - 3} = \frac{0}{-6} = 0.$$
The denominator is nonzero at every step, so the value is defined and equals $0$.

The answer is $\boxed{\textbf{(C)}\ 0}$.