Alright, let's solve this problem step by step. We have 1141 and 6677 and we're adding them together.

Step 1: We'll start by adding the digits 1 & 7 in column 1 and get 8.

Step 2: We'll start by adding the digits 4 & 7 in column 2 and get 11.
We'll write down the last digit 1 and carry the 1 to the next column.

Step 3: We'll start by adding the digits 1 & 6 in column 3 and get 8.

Step 4: We'll start by adding the digits 1 & 6 in column 4 and get 7.

So, 1141 + 6677 = 7818.

