Problem 36 - Fundamentals of Mathematics

Question

For the following problems, perform the subtractions. You may chedk each difference with a calculator. $$ \begin{array}{r} 35,002 \\ -14,001 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The result of the subtraction is 21,001.

Step 1: Identify Each Digit for Subtraction

Align the numbers vertically by their place values. Starting from the rightmost digit (the units column), identify the digits that need to be subtracted: 2 - 1 in the units place, 0 - 0 in the tens place, 0 - 0 in the hundreds place, 5 - 4 in the thousands place, and 3 - 1 in the ten-thousands place.

Step 2: Subtract Units Place

Begin with the rightmost column, the units place. Subtract the bottom digit from the top digit: 2 - 1 = 1.

Step 3: Subtract Tens Place

Move to the tens place. Subtract the bottom digit from the top digit: 0 - 0 = 0.

Step 4: Subtract Hundreds Place

Move to the hundreds place. Subtract the bottom digit from the top digit: 0 - 0 = 0.

Step 5: Subtract Thousands Place

Move to the thousands place. Subtract the bottom digit from the top digit: 5 - 4 = 1.

Step 6: Subtract Ten-Thousands Place

Move to the ten-thousands place. Subtract the bottom digit from the top digit: 3 - 1 = 2.

Step 7: Combine Results

Combine the digits obtained from each subtraction step: 2 (ten-thousands), 1 (thousands), 0 (hundreds), 0 (tens), and 1 (units). Thus, the result is 21,001.

Key Concepts

Place Value

Understanding place value is fundamental to performing any arithmetic operation, especially subtraction. Each digit in a number holds a specific position, which determines its value. For instance, in the number 35,002:

• The '2' is in the units place, meaning it represents 2 one-dollar bills.
• The '0' in the tens place means zero tens or groups of ten (10).
• The next '0' in the hundreds place also represents zero but this time as groups of one hundred (100).
• The '5' is located in the thousands place, which equates to five groups of one thousand (1,000).
• Finally, the '3' in the ten-thousands place means three groups of ten thousand (10,000).

We start at the rightmost digit when subtracting to ensure each digit is correctly aligned according to its value. By understanding this, we can effectively navigate through subtraction, ensuring accuracy and precision in arithmetic operations.

Borrowing in Subtraction

Borrowing, also known as regrouping, is an essential technique when dealing with subtraction where a given place value does not have enough to subtract from. Imagine subtracting 7 from 3! We begin by borrowing from the next higher place value. For example, if we need to subtract 14,001 from 35,002:

• If we encounter a situation where a smaller digit is on top, such as subtracting a larger bottom digit from it, we'd check the next higher place value to 'borrow' from it.
• Through borrowing, we convert a '10' from the next left place value to handle the subtraction. Essentially, this process redistributes part of the value to the right column, enhancing our ability to subtract efficiently.

Using borrowing simplifies complex subtraction problems, converting potential obstacles into solvable arithmetic. In the exercise given, borrowing isn't required, but it is vital for cases where it becomes necessary.

Arithmetic Operations

Arithmetic operations like addition, subtraction, multiplication, and division form the basis of mathematics. Subtraction, specifically, involves finding the difference between two numbers. In our example: - We aligned 35,002 and 14,001 vertically, ensuring each digit lined up with the corresponding place value for accurate subtraction. - Starting from the right, we conducted subtraction for each digit individually. First 2-1 for the units, 0-0 for tens, 0-0 for hundreds, 5-4 for thousands, and 3-1 for ten-thousands. Each operation removes a specific amount, making subtraction essential in calculations such as determining remaining quantities, calculating distances, or even financial transactions. Mastering each part of this arithmetic operation, given its everyday applications, thereby strengthens one's overall numerical proficiency.