Problem 42 - Fundamentals of Mathematics

Question

Combine the numbers as indicated. $$350,212+14,533$$

Step-by-Step Solution

Answer

The sum is 364,745.

Step 1: Align the Numbers

First, write the numbers one on top of the other, aligning them by their rightmost digits:\[\begin{array}{c} 350,212 \+ 14,533 \\hline\end{array}\]

Step 2: Add Units Column

Add the digits in the units column: \(2 + 3 = 5\). Write 5 below the line, under the units column.

Step 3: Add Tens Column

Add the digits in the tens column: \(1 + 3 = 4\). Write 4 in the tens column.

Step 4: Add Hundreds Column

Add the digits in the hundreds column: \(2 + 5 = 7\). Write 7 in the hundreds column.

Step 5: Add Thousands Column

Add the digits in the thousands column: \(0 + 4 = 4\). Write 4 in the thousands column.

Step 6: Add Ten-Thousands Column

Add the digits in the ten-thousands column: \(5 + 1 = 6\). Write 6 in the ten-thousands column.

Step 7: Add Hundred-Thousands Column

The number 350,212 has a digit in the hundred-thousand column, but 14,533 does not. So, just bring down the digit from 350,212: 3.

Step 8: Write Final Answer

Combine all of the added columns together to get the final result: 364,745. So:\[\begin{array}{c} 350,212 \+ 14,533 \\hline 364,745\\end{array}\]

Key Concepts

Place Value Alignment

When performing addition on large numbers, it's crucial to ensure correct place value alignment. This means lining up the numbers so that the digits of the same value are directly above each other. For instance, hundreds should be above hundreds, units above units, and so on. This helps in accurately performing the operation and prevents errors.

To achieve this, start from the rightmost digit of each number, which represents the units. Align these digits vertically in a column. Then, align the next left digit that represents tens, and continue this process for hundreds, thousands, and beyond.

Proper alignment ensures that you are adding each part of the numbers correctly and efficiently. If the numbers aren't aligned properly, it could lead to incorrect summation.

Column-Wise Addition

Once you've aligned the digits based on their place values, you can begin adding them column by column. This method is known as column-wise addition. By working one column at a time, it simplifies the process and helps to focus on individual building blocks of the numbers.

Start from the rightmost column, the units. Add the numbers in this column first. For example, if in your units column you have a 2 and a 3, then you sum them to get 5. Write this sum beneath the line in the same column.

Move to the left to the tens column. Add these digits together and write down the result just like with the units. Repeat this process for each column to the left until all columns are added. This systematic approach reduces errors and keeps the process organized.

Carry Over in Addition

In some instances, especially when adding digits of 9, you might need to carry over in addition. This happens when the sum of a column's digits is greater than 9. You will carry over the extra value to the next column to the left.

For example, if you're adding 7 and 5, you get 12. Write down 2 in the current column and carry over the 1 (from 12) to the next left column. Add this carry-over number with the digits of that column.

This method ensures that your results are accurate by accounting for every additional value. It's similar to when you think of how the number 10 is actually 0 in units place and 1 in tens place. Carrying over aligns with breaking down the number into its constituents.