Problem 12 - Fundamentals of Mathematics

Question

Find the sums and differences. $$ \begin{array}{r} 80,349 \\ +\quad 2,679 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum is 83,028.

Step 1: Line Up the Numbers

Align the numbers so that their decimals or their place values are in a straight line. This alignment ensures that each place value is correctly added. Here, write 80,349 and align +2,679 such that both numbers are right-aligned.

Step 2: Add the Rightmost Digits

Start by adding the rightmost digits (ones place). Here, add 9 (from 80,349) and 9 (from 2,679) which sums up to 18. Write down 8 and carry over 1 to the next column.

Step 3: Proceed with the Tens Digit

Next, add the tens place. Add 4 (from 80,349), 7 (from 2,679), and the carried over 1. This equals 12. Write down 2 and carry over 1 to the next column.

Step 4: Move to the Hundreds Digit

Add the hundreds place. Add 3 (from 80,349), 6 (from 2,679), and the carried over 1. The result is 10. Write down 0 and carry over 1.

Step 5: Add the Thousands Digit

Now, add the thousands place. Here, add 0 (from 80,349), 2 (from 2,679), and the carried over 1. This gives 3, which you write down.

Step 6: Add the Ten Thousands Digit

Add the ten thousands. Here, you just have 8 since the other number does not have a digit in this place value. Thus, you write 8.

Step 7: Write Down the Final Result

Now that all digits have been added, combine all parts from previous steps to write the final sum, which is 83,028.

Key Concepts

Place Value

Place value is a core concept in mathematics, especially useful when performing operations like addition and subtraction. It helps us understand the value of each digit in a number, based on its position. Let's break this down:

• The rightmost digit is in the 'ones' place, representing units.
• The next digit to the left is the 'tens' place, and it represents ten times the value of the 'ones'.
• This pattern continues with 'hundreds', 'thousands', and so on, each being ten times the value of the previous place.

Understanding the place value system is essential because each digit contributes differently to the overall value of the number. When numbers are aligned by their place values, it becomes easier to add or subtract them without confusion.

Carrying Method

The carrying method is an important technique in addition, especially when the sum of digits exceeds nine. Here's how it works:

• When the sum of digits in a place value column is ten or more, you write down the digit in the 'ones' place, and carry over the digit in the 'tens' place to the next column on the left.
• For example, if two digits add up to 18, you write down 8 and carry over 1 to the next higher place value.
• This ensures accurate addition across all columns, preserving the values of larger digits.

Carrying is crucial to getting the correct result in addition and demonstrates the significance of understanding place value in this process.

Column Addition

Column addition is a methodical way of adding numbers. It involves lining up numbers vertically according to their place values, as seen in the given problem. Here’s a brief step-by-step guide:

• Align the numbers: Ensure all digits are positioned in their corresponding place value columns.
• Start from the right: Begin adding the numbers from the rightmost column (ones place) to the left, one column at a time.
• Use carrying method: If necessary, carry over extra values to the next column.

This strategy simplifies complex addition problems, making it easier to visualize and understand the process of adding large numbers, like in our exercise. Column addition bridges abstract mathematical operations with practical, understandable steps for students.

Arithmetic

Arithmetic is the branch of mathematics dealing with numbers and basic operations: addition, subtraction, multiplication, and division. In this exercise, we are focused on addition, which is a fundamental arithmetic operation. Let's look at why arithmetic is important:

• It provides the foundational skills needed for more advanced mathematical concepts.
• Understanding arithmetic operations helps solve everyday problems, from counting change to calculating expenses.
• Through arithmetic, students learn to approach problems logically and develop analytical thinking.

In this specific case, using arithmetic, the sum of 80,349 and 2,679 was found using addition principles, demonstrating the practical application of these basic skills. It shows how arithmetic is not just theoretical knowledge but a tool we use in everyday life.