Problem 10 - Fundamentals of Mathematics

Question

For the next three problems, find the sums. $$ \begin{array}{r} 16,945 \\ 8,472 \\ 387,721 \\ 21,059 \\ 629 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum is 434,826.

Step 1: Align the numbers vertically

First, write down the numbers to add in a column format for better clarity. Make sure each number is aligned by the rightmost digit, like this:\[\begin{array}{r}16,945 \8,472 \387,721 \21,059 \629 \\hline\end{array}\]

Step 2: Add the units place

Start adding the numbers from the rightmost column, which is the units place (or ones).\[5 + 2 + 1 + 9 + 9 = 26\]Write 6 in the units column of the answer and carry over 2 to the tens column.

Step 3: Add the tens place

Next, add the numbers in the tens place along with the carry-over.\[4 + 7 + 2 + 5 + 2 + 2 = 22\]Write 2 in the tens column of the answer and carry over 2 to the hundreds column.

Step 4: Add the hundreds place

Add the numbers in the hundreds place along with the carry-over.\[9 + 4 + 7 + 0 + 6 + 2 = 28\]Write 8 in the hundreds column of the answer and carry over 2 to the thousands column.

Step 5: Add the thousands place

Add the numbers in the thousands place with the carry-over.\[6 + 8 + 7 + 1 + 2 = 24\]Write 4 in the thousands column of the answer and carry over 2 to the ten thousands column.

Step 6: Add the ten thousands place

Add the numbers in the ten thousands place along with the carry-over.\[1 + 3 + 2 = 6\]Write 6 in the ten thousands column of the answer. Since there are no more digits to add, finalize by writing:\[\boxed{434,826}\]Hence, the final sum is 434,826.

Key Concepts

Place Value

Understanding place value is crucial in mathematics, especially when performing operations like addition. Each digit in a number holds a different value based on its position or 'place' in the number. This can mean the difference between a number being worth ten, one hundred, or one thousand depending on its place.
For instance, in the number 16,945:

• 5 is in the units (ones) place, meaning it represents 5.
• 4 is in the tens place, indicating it represents 40.
• 9 is in the hundreds place, so it stands for 900.
• 6 is in the thousands place, making it 6,000.
• 1 is in the ten thousands place, representing 10,000.

Each of these positions increases in value by a factor of ten as you move from right to left. Understanding this positional notation not only helps in addition but is the foundation for all arithmetic operations.

Column Addition

Column addition is a method used to add numbers together by writing them one under the other. This is done in such a way that each digit aligns with its corresponding place value. The numbers are usually aligned to the rightmost digits, which helps keep the addition organized and clear.
Here’s how you do it:

• Write the numbers vertically, making sure each digit lines up with its counterparts according to place value.
• Begin adding from the rightmost column, which is the units place.
• Move sequentially leftward to the tens, hundreds, thousands, and so on.

This technique is particularly useful for handling numbers of any size and ensures accuracy as each digit is added separately, accounting for any carry over from previous columns.

Carry Over

Carry over is an important concept when digits in a column sum to a number greater than 9. In such cases, the sum is split into two parts: the digit in the current column and a carry-over to the next column.

For example, consider adding 5 + 7, which equals 12:

• You would write 2 in the column you're adding and carry over 1 to the next column to the left, often the tens column.

This carrying process continues from the right to the leftmost digit. In larger sums, it may require carrying over through several consecutive columns if the sums continue to exceed 9.

This means after adding the units, tens, and lastly hundreds places, you ensure each column correctly contributes to the total sum, reducing errors and reinforcing understanding of number value distribution.