Calculate the Sum: 2781 + 8417 Using Vertical Addition

$\begin{aligned} &2781 \\ +& \\ &8417 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

To solve this problem, we'll follow these steps:

• Step 1: Set up the numbers for addition in a vertical format, aligning the rightmost digits.
• Step 2: Add each column starting from the right, keeping track of the carryover.
• Step 3: Continue the addition for all columns, incorporating any carryover into the next left column.

Now, let's work through the addition process:

Add the units column: $1 + 7 = 8$. No carryover occurs here.

Next, add the tens column: $8 + 1 = 9$. Still, no carryover is involved.

Then, add the hundreds column: $7 + 4 = 11$. Because this is more than 9, we write down 1 and carry over 1 to the next column.

Finally, add the thousands column: $2 + 8 = 10$, plus the 1 carried over gives 11. Thus, we write down 11.

Therefore, the sum of the numbers is $\boxed{11198}$.