Suppose that $3\triangle_4=\triangle2_{11}$, where $\triangle$ represents a digit 0-9. What is the value of the missing digit?
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. 