Problem 38 - Fundamentals of Mathematics

Question

Find the sum of $$ \begin{array}{r} 482 \\ +\quad 68 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum is 550.

Step 1: Align the Numbers

Write the two numbers vertically, aligning them by their rightmost digits. So, it will be \(482\) directly above \(68\).

Step 2: Add the Units Column

Add the digits in the units column: \(2 + 8 = 10\). Write down \(0\) and carry over \(1\) to the tens column.

Step 3: Add the Tens Column

Add the digits in the tens column including the carry-over: \(8 + 6 + 1 = 15\). Write down \(5\) and carry over \(1\) to the hundreds column.

Step 4: Add the Hundreds Column

Add the digits in the hundreds column including the carry-over: \(4 + 1 = 5\). Write down \(5\).

Step 5: Write the Result

Combine all the digits from each column: the hundreds \(5\), tens \(5\), and units \(0\), to get the final sum of \(550\).

Key Concepts

Aligning Numbers

Aligning numbers is an essential step in addition when dealing with multi-digit numbers. It sets the foundation for adding numbers correctly by placing each digit in the right column based on its place value.

• Start by writing the numbers to be added vertically, ensuring that the rightmost digits are aligned with each other.
• Draw a line under the numbers to set a clear visual separation for the work below.
• Be attentive to each digit's "neighborhood," meaning its position in the number (units, tens, hundreds, etc.).

By correctly aligning numbers, mistakes related to erroneous `place value` calculations are minimized. Proper alignment is the first crucial step in mastering addition.

Carrying Over

Carrying over is a fundamental technique used in addition when a sum in a column exceeds what can be held in that column. It ensures that extra value is efficiently transferred to the next significant column.

• When the sum of digits in any column is 10 or more, write the unit digit in that column.
• The leftmost digit, often called the 'carry', is transferred to the adjacent higher column.
• This process repeats as needed through each column of digits.

For example, adding the units place in the initial example resulted in 10, so the 0 was written down, and the 1 was carried over to the tens place. This meticulous process helps to keep addition organized and accurate.

Place Value

Understanding place value is key to correctly performing arithmetic operations like addition. Each digit in a number has a value based on its position or place within the number.

• The rightmost position is the units place.
• The next column to the left is the tens place, where each digit represents ten times its face value.
• After that comes the hundreds place, and similarly, each digit represents a hundred times its face value.

Recognizing and utilizing place value allows you to align numbers properly and carry over correctly, ensuring precision in calculations. This concept is foundational and extends beyond addition to all arithmetic operations.

Step by Step Solution

Breaking down the addition process into simple steps helps in clearly understanding and executing the arithmetic.

• Step 1: Start by aligning the numbers correctly and identifying the place values.
• Step 2: Add the digits in the units column, carefully noting any carry over.
• Step 3: Move to the tens column, adding any carried value and digits, checking once more for another potential carry.
• Step 4: Lastly, add the hundreds column, incorporating any carry overs.
• Step 5: Combine the results of all columns to find the final sum.

Each step ensures that all aspects of the addition, from placement to final calculation, are methodical and accurate. Following these steps can lead to the successful completion of even more complex addition problems.