Okay, we are tasked with adding 8255 and 9525. Let's begin!

Step 1: We'll start by adding the digits 5 & 5 in column 1 and get 10.
We'll write down the last digit 0 and carry the 1 to the next column.

Step 2: We'll start by adding the digits 5 & 2 in column 2 and get 8.

Step 3: We'll start by adding the digits 2 & 5 in column 3 and get 7.

Step 4: We'll start by adding the digits 8 & 9 in column 4 and get 17.
We'll write down the last digit 7 and carry the 1 to the next column.

So, 8255 + 9525 = 17780.

