For each pair of real numbers a ≠ b, define the operation \star as (a \star b) = (a+b)/(a-b). What is the value of ((1 \star 2) \star 3)?

2005 AMC 10A Problem 2 — Solution

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For each pair of real numbers $a \neq b,$ define the operation $\star$ as $(a \star b) = \frac{a+b}{a-b}.$

What is the value of $((1 \star 2) \star 3)?$

$-\dfrac{2}{3}$

$-\dfrac{1}{5}$

$0$

$\dfrac{1}{2}$

This value is not defined.

First $(1 \star 2) = \dfrac{1+2}{1-2} = \dfrac{3}{-1} = -3.$ Then $(-3 \star 3) = \dfrac{-3+3}{-3-3} = \dfrac{0}{-6} = 0.$

Thus, the correct answer is C.