### Instruction: Suppose that $3\triangle_4=\triangle2_{11}$, where $\triangle$ represents a digit 0-9. What is the value of the missing digit?### Response: To solve this problem, I need to use the place value of each digit and the base of each numeral system.
In base 4, each digit represents a power of 4, so $3\triangle_4=3\times 4^1+\triangle\times 4^0$.
In base 11, each digit represents a power of 11, so $\triangle2_{11}=\triangle\times 11^1+2\times 11^0$.
Since these two expressions are equal, I can set them equal to each other and simplify: $3\times 4^1+\triangle\times 4^0=\triangle\times 11^1+2\times 11^0$.
Expanding the powers, I get $12+\triangle=11\triangle+2$.
Subtracting 2 from both sides, I get $10+\triangle=11\triangle$.
Subtracting $\triangle$ from both sides, I get $10=10\triangle$.
Dividing both sides by 10, I get $1=\triangle$.
So the missing digit is 1.