Problem 35 - Fundamentals of Mathematics

Question

Find the sums and differences. $$ \begin{array}{r} 676,504 \\ -\quad 58,277 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

618,227

Step 1 - Understand the Problem

We are given two numbers, 676,504 and 58,277. We need to find the result of subtracting 58,277 from 676,504.

Step 2 - Arrange the Numbers

Write the numbers in a vertical line, ensuring that the larger number (676,504) is on top, and align the digits according to their place values.

Step 3 - Subtract from Right to Left

Start subtracting from the rightmost digit (units place). If you cannot subtract because the top digit is smaller, borrow 1 from the next left digit.

Step 4 - Subtract the Units Place

4 (units) subtracted by 7. Since 4 is smaller than 7, we borrow 1 from the tens place, making it 14 - 7 = 7. The tens place now becomes 9 (because we borrowed 1).

Step 5 - Subtract the Tens Place

9 (tens) subtracted by 7 equals 2 since we borrowed 1 earlier reducing the top number's tens from 10 to 9.

Step 6 - Subtract the Hundreds Place

5 (hundreds) subtracted by 2 equals 3.

Step 7 - Subtract the Thousands Place

6 (thousands) subtracted by 8. Since 6 is smaller, we borrow from the ten-thousands. 16 - 8 = 8. The ten-thousands place will now be 5.

Step 8 - Subtract the Ten-Thousands Place

5 (ten-thousands) subtracted by 5 equals 0.

Step 9 - Subtract the Hundred-Thousands Place

6 (hundred-thousands) minus nothing remains 6.

Step 10 - Assemble the Answer

Combine all results from each place value to get the final answer: 618,227.

Key Concepts

Place Value

In subtraction, understanding place value is crucial. Place value refers to the position of a digit in a number, which determines its actual value. Each digit's place—from units, to tens, hundreds, and so on—represents a power of 10. For instance:

• The rightmost digit is the "ones" or "units" place, valued at 10⁰.
• Next is the "tens" place, valued at 10¹.
• The "hundreds" place is 10², and so forth.

Understanding place value helps ensure each digit is subtracted from the correct matching digit in the same column. This is particularly important when performing subtraction or any other math operations.

Borrowing in Subtraction

Borrowing in subtraction, also known as "regrouping," is a technique used when a digit in the top number is smaller than the corresponding digit in the bottom number. Borrowing allows us to temporarily adjust the numbers to perform the subtraction. For example, when subtracting 7 from 4, we borrow 1 from the next left place value:

• The 4 becomes 14, allowing the subtraction of 7, resulting in 7.
• The tens place, from which we borrowed, is reduced by 1.

Remember that borrowing is not taking away permanently; it's a momentary adjustment to facilitate subtraction.

Mathematics Problem Solving

Mathematics problem solving involves breaking down the problem into smaller, manageable parts. For this subtraction:

• Understanding the goal is crucial. Here, the task is to find the difference between two numbers.
• Identify and apply necessary techniques such as borrowing when needed.
• Accuracy is key, so always recheck each step to ensure no miscalculation.

Problem solving in mathematics is fundamentally about understanding the problem first and applying appropriate methods logically, step-by-step.

Vertical Alignment in Math Operations

When working with multi-digit numbers, organizing them into vertical alignment is highly important. This arrangement ensures that you are working with each digit in its appropriate place value column:

• The larger number should be on top; the numbers must be aligned by place value.
• This alignment helps in borrowing and carrying operations, guaranteeing accuracy.
• It simplifies the subtraction process by making it clear which numbers to subtract from one another in sequential order.

By using vertical alignment, you minimize errors and see the flow of numbers more clearly, which is especially helpful in lengthy arithmetic calculations.