The variable y varies inversely with x. When x=3, y=8. What is the value of y when x=12?

Inverse variation means the product $xy$ is constant. From $x = 3$ and $y = 8$, the constant is $k = 3 \cdot 8 = 24$. When $x = 12$, $y = \frac{24}{12} = 2$. Sanity check: $x$ was multiplied by 4, so $y$ should be divided by 4, and $8 \div 4 = 2$. The most common wrong answer is 32, which comes from treating the relationship as direct variation ($y$ multiplied by 4 along with $x$). Another wrong answer is 20, from subtracting the difference in $x$ (9) incorrectly. Inverse variation always involves dividing, not multiplying.