First, the word 'appreciated' means increased in value. If a house appreciates in value by 20%, it is now worth 20% more than its original price.

Let us call the original value of the rug \(x\). We know that the rug is worth 105% of what it was worth 10 years ago. We can write this as

\(12000 = 105\% \cdot x\). Converting the percent to a decimal gives us the equation \(12000=1.05x\). Dividing both sides by 1.05 gives us our final answer: \(x \approx \$11428.57\). (Rounded to the nearest cent.) So, the antique rug was worth approximately $11428.57.