### Instruction: Each of the symbols $\star$ and $*$ represents an operation in the set $\{+,-,\times,\div\}$, and $\frac{12\star 2}{9*3}=2$.  What is the value of $\frac{7\star 3}{12*6}$?  Express your answer as a common fraction.### Response: I notice that the given equation involves two operations and four numbers, and I need to find two operations that make it true.
One way to do that is to try different combinations of operations and see which ones work.
I start with the simplest operations, addition and subtraction, and try them for $\star$ and $*$.
If $\star$ is addition, then $12\star 2=14$, and I need to find an operation $*$ that makes $\frac{14}{9*3}=2$.
That means $9*3$ must be $7$, which is impossible with any of the given operations.
So $\star$ cannot be addition.
If $\star$ is subtraction, then $12\star 2=10$, and I need to find an operation $*$ that makes $\frac{10}{9*3}=2$.
That means $9*3$ must be $5$, which is also impossible with any of the given operations.
So $\star$ cannot be subtraction either.
That leaves multiplication and division for $\star$.
If $\star$ is multiplication, then $12\star 2=24$, and I need to find an operation $*$ that makes $\frac{24}{9*3}=2$.
That means $9*3$ must be $12$, which is possible if $*$ is division.
So one possible solution is $\star=\times$ and $*=\div$.
To check, I plug these operations into the expression $\frac{7\star 3}{12*6}$ and see what I get.
I get $\frac{7\times 3}{12\div 6}=\frac{21}{2}$, which is a common fraction.
This could be the answer, but I need to make sure there are no other possible solutions.
So I try the other case, where $\star$ is division.
If $\star$ is division, then $12\star 2=\frac{12}{2}=6$, and I need to find an operation $*$ that makes $\frac{6}{9*3}=2$.
That means $9*3$ must be $3$, which is impossible with any of the given operations.
So $\star$ cannot be division.
Therefore, the only possible solution is $\star=\times$ and $*=\div$.
So the value of $\frac{7\star 3}{12*6}$ is $\frac{21}{2}$.