To add the binary numbers $111.11$ and $101.10$, we perform the addition column by column from right to left, following the rules of binary arithmetic ($0+0=0, 0+1=1, 1+0=1, 1+1=10$): $$ \begin{array}{r@{\quad}l} 111.11_2 & \\ + 101.10_2 & \\ \hline 1101.01_2 & \end{array} $$ Step-by-step calculation: 1. In the rightmost column (hundredths place): $1 + 0 = 1$. 2. In the next column (tenths place): $1 + 1 = 10$. Write $0$ and carry over $1$. 3. In the units column: $1 + 1 + 1 \text{ (carry)} = 11$. Write $1$ and carry over $1$. 4. In the twos column: $1 + 0 + 1 \text{ (carry)} = 10$. Write $0$ and carry over $1$. 5. In the fours column: $1 + 1 + 1 \text{ (carry)} = 11$. Write $11$. Therefore, the sum is $(1101.01)_2$.