### Instruction: Express $1.\overline{234}$ as an improper fraction.### Response: I notice that the decimal part of the number is repeating, so I can use a trick to get rid of it.
I let x = 1.\overline{234} and multiply both sides by 1000, since there are three digits in the repeating part.
This gives me 1000x = 1234.\overline{234}.
Now I can subtract x from both sides to eliminate the decimal part.
This gives me 999x = 1233, which is a much simpler equation.
I can divide both sides by 3 to simplify it further.
This gives me 333x = 411.
Finally, I can divide both sides by 333 to get x by itself.
This gives me x = 411/333.
So, the answer is 1.\overline{234} = 411/333.