Problem 39 - Fundamentals of Mathematics

Question

Add the numbers. $$ \begin{array}{r} 2,129 \\ 6,190 \\ 17,044 \\ 30,447 \\ 292 \\ 41 \\ 428,458 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum is 484,601.

Step 1: Align the Numbers

Align all the numbers vertically by their least significant digit (rightmost digit). This helps in ensuring that each column contains digits of the same place value (units, tens, hundreds, etc.).

Step 2: Add Units Column

Start by adding the digits in the units column (rightmost column): 9 + 0 + 4 + 7 + 2 + 1 + 8 = 31. Write down 1 and carry over 3 to the tens column.

Step 3: Add Tens Column

Add the digits in the tens column, including the carryover from the previous step: 2 + 9 + 4 + 4 + 9 + 4 + 5 + 3 (carryover) = 40. Write down 0 and carry over 4 to the hundreds column.

Step 4: Add Hundreds Column

Add the digits in the hundreds column, including the carryover: 1 + 1 + 0 + 4 + 2 + 0 + 4 + 4 (carryover) = 16. Write down 6 and carry over 1 to the thousands column.

Step 5: Add Thousands Column

Add the digits in the thousands column, accounting for the carryover: 2 + 6 + 7 + 0 + 0 + 0 + 8 + 1 (carryover) = 24. Write down 4 and carry over 2 to the ten-thousands column.

Step 6: Add Ten-Thousands Column

Add the digits in the ten-thousands column, including the carryover: 0 + 0 + 1 + 3 + 0 + 0 + 2 (carryover) = 6. Write down 6.

Step 7: Add Hundred-Thousands Column

Add the digit in the hundred-thousands column: 4. Write down 4, as there is no carryover or other number to add in this column.

Key Concepts

Place Value

Understanding place value is essential when working with addition problems, especially with large numbers like in our original exercise. The place value system helps us know what each digit in a number represents, based on its position. In the number 2,129, for example:

• The digit 9 is in the 'units' place, so it represents 9 ones.
• The digit 2 is in the 'tens' place, representing 20.
• The digit 1 is in the 'hundreds' place, representing 100.
• The digit 2 is in the 'thousands' place, representing 2,000.

Recognizing these positions is crucial because it allows us to align numbers correctly for addition. Each column we add corresponds to a specific place value, ensuring that we are combining numbers correctly from similar categories, such as units with units or tens with tens. This lays the groundwork for accurate mathematical calculations and helps in preventing errors.

Column Addition

Column addition is a method used to add numbers stacked on top of each other, ensuring they are aligned correctly by place value. This method simplifies adding long lists of numbers like in our exercise.
The process works as follows:

• Step 1: Write each number in a column, ensuring that digits align by their place values (units under units, tens under tens, etc.).
• Step 2: Begin additions from the rightmost column (usually the units column) and work your way left.
• Step 3: Add the digits in each column, writing the result below, in their respective position.

Column addition helps you visualize the addition process and ensures each place value is handled separately, reducing the chances of missing any numbers. It allows for easy identification of where to carry over numbers if needed, which brings us to the next concept.

Carrying in Addition

Carrying in addition comes into play when the sum in a column reaches double digits, exceeding the place value. Here's how it works in detailed steps:

• When you add numbers in a column and the result is 10 or more, write down the rightmost digit (units), and carry over the left digit (or digits) to the next column on the left.
• For instance, if a column sums up to 31, you write down 1 and carry over 3 to the next column.
• Include the carried number in your addition of the next column, treating it as an extra addend.

Carrying helps manage large sums efficiently across multiple columns and ensures that each position value is respected. It's important to consistently apply this strategy to achieve correct results, just as seen in the steps of the exercise where carrying is managed at each stage to accurately compute the total sum.