How many unique values can be created by forming the fraction $\frac{x}{y}$ where $x$ is either 4, 8, or 12 and $y$ is either 4, 8, or 12?
That means I need to simplify each fraction and check for duplicates. I also notice that all the possible values of $x$ and $y$ are multiples of 4, so I can divide both the numerator and the denominator by 4 to simplify them. For example, $\frac{4}{4} = \frac{1}{1} = 1$, $\frac{4}{8} = \frac{1}{2}$, and $\frac{4}{12} = \frac{1}{3}$. If $x = 8$, then $\frac{8}{4} = \frac{2}{1} = 2$, $\frac{8}{8} = \frac{1}{1} = 1$, and $\frac{8}{12} = \frac{2}{3}$. If $x = 12$, then $\frac{12}{4} = \frac{3}{1} = 3$, $\frac{12}{8} = \frac{3}{2}$, and $\frac{12}{12} = \frac{1}{1} = 1$. Now I have nine simplified fractions, but some of them are duplicates. I see that $1$ appears three times, $2$ appears once, $3$ appears once, $\frac{1}{2}$ appears once, $\frac{1}{3}$ appears once, $\frac{2}{3}$ appears once, and $\frac{3}{2}$ appears once. So there are only seven unique values that can be created by forming the fraction $\frac{x}{y}$ where $x$ is either 4, 8, or 12 and $y$ is either 4, 8, or 12. 