Problem 42 - Fundamentals of Mathematics

Question

For the following problems, perform the additions. If you can, check each sum with a calculator. $$ \begin{array}{r} 631,464 \\ +509,740 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum is 1,141,204.

Step 1: Align the Numbers

Write down the numbers 631,464 and 509,740 such that their digits are aligned vertically by place value: units under units, tens under tens, etc.

Step 2: Add the Units Column

Start with the units column (rightmost side): 4 + 0 = 4. Write 4 below the line under the units column.

Step 3: Add the Tens Column

Proceed to the tens column: 6 + 4 = 10. Write 0 below the line under the tens place and carry over the 1 to the next column.

Step 4: Add the Hundreds Column

Move to the hundreds column: 4 + 7 + 1 (carry over) = 12. Write 2 below the line under the hundreds place and carry over the 1.

Step 5: Add the Thousands Column

Add the thousands column: 1 + 9 + 1 (carry over) = 11. Write 1 below the line under the thousands place and carry over the 1.

Step 6: Add the Ten Thousands Column

Now proceed to the ten thousands column: 3 + 0 + 1 (carry over) = 4. Write 4 below the line under the ten thousands place.

Step 7: Add the Hundreds of Thousands Column

Finally, add the hundreds of thousands column: 6 + 5 = 11. Write 11 entirely below the line because it's the final sum.

Key Concepts

Carrying in Addition

Carrying in addition is a fundamental concept that helps in adding numbers where the sum of a column exceeds 9. This process ensures each column in the addition is correctly calculated by carrying over excess value to the next column.
Let's break down the steps:

• Each column represents a different place value, such as units, tens, hundreds, etc.
• When the digits in a column add up to a two-digit number, the leftmost digit is the 'carry over' to the next column.
• In the addition of 631,464 and 509,740, the tens column sums to 10. Here, '0' is written in the tens place while '1' is carried over to the hundreds column.

This carry over increases the total in the next column, and it's crucial to keep track of these values to achieve the correct total sum.

Column Addition

Column addition is a straightforward way of adding numbers that aligns each digit according to its place value. This traditional method simplifies the addition of large numbers.
Here's how to perform column addition step-by-step:

• Align each number vertically, ensuring that units are under units, tens under tens, and so on. This alignment ensures the correct addition of corresponding place values.
• Start adding from the rightmost column, which is the units column. Work from the bottom to the top of each column, recording the sum or part of it below the column.
• If the sum of any column exceeds 9, use carrying to move to the next column.
• Continue adding column by column until all have been processed, always keeping an eye out for possible carry overs.

This method is effective and ensures that large sums are broken down into manageable steps. It is particularly useful when handling multi-digit numbers as in our exercise example.

Check Sums With Calculator

Checking sums with a calculator is a great way to ensure the accuracy of your manually calculated results. A calculator provides a quick and reliable confirmation of your work.
When you solve a problem using column addition or any other method, a calculator serves as an efficient tool to:

• Quickly verify sums for larger numbers, which reduces the chance of errors.
• Get immediate feedback, confirming whether the manual calculations were performed correctly.
• Help identify any mistakes in carrying or initial addition operations.

Using a calculator after you've completed manual addition will bolster confidence in your numerical skills and verify the integrity of your arithmetic processes. Besides, it ensures that your answer is both correct and dual-verified by two methods, adding valuable experience to your understanding of the topic.