Problem 52 - Fundamentals of Mathematics

Question

For the following problems, perform the subtractions. You may chedk each difference with a calculator. $$ \begin{array}{r} 10,113 \\ -\quad 2,079 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The difference is 8,034.

Step 1: Write down the problem

Start by writing the numbers in a vertical format, aligning by the right.\[\begin{array}{r}10,113 \-\quad 2,079 \\hline\end{array}\]

Step 2: Subtract the units column

Look at the units column (rightmost digit). Subtract 9 from 3. Since 3 is smaller than 9, you need to borrow from the tens column. Change the 1 in the tens column to 0 and the 3 in the units column to 13. Now, 13 - 9 = 4.

Step 3: Subtract the tens column

Move to the tens place. After borrowing, subtract 7 from 0. Since you can't subtract 7 from 0, borrow 1 from the hundreds column. The hundreds column becomes 0, and the tens column becomes 10. Now, 10 - 7 = 3.

Step 4: Subtract the hundreds column

After borrowing, subtract 0 from 0. The result is 0 because 0 minus 0 is 0.

Step 5: Subtract the thousands column

Subtract 2 from 0. Since you cannot subtract 2 from 0, borrow 1 from the ten thousands column. The ten thousands column becomes 9, and the thousands column becomes 10. Now, 10 - 2 = 8.

Step 6: Write down the final result

Combine all the columns to state the final difference, which is 8,034.

Key Concepts

Borrowing in Subtraction

Borrowing in subtraction is a powerful tool that helps us process numbers when we encounter bigger numbers to subtract from smaller ones. This often occurs in columns when performing vertical subtraction. Borrowing is essentially about redistributing numbers from the higher place value.

It involves adjusting the figures to accommodate the subtraction:

• If a digit is smaller than the number you are subtracting, borrow from the next higher digit to the left.
• Each borrowing reduces the left-side digit by 1 and increases your current digit by 10.
• This method ensures that each subtraction is possible, even when the top number is smaller.

In our exercise, borrowing was used multiple times to accomplish subtraction effectively:

• First, from the tens to the units column, changing 3 to 13.
• Next, from the hundreds to the tens, changing 0 to 10.
• Lastly, from the ten thousands to the thousands, changing 0 to 10.

By following these borrowing steps, subtraction becomes feasible, even for seemingly difficult numbers.

Place Value

Place value is the system that helps us understand the value of each digit in a number based on its position. Each position in a number has a specific value, starting from units, tens, hundreds, and so on.

Understanding place value is crucial in mathematical operations like subtraction, as each column represents a different power of ten:

• Units (the rightmost digits) represent the number itself, without multiplication.
• The tens place is ten times the value in the units place.
• The hundreds place is a hundred times the units place value.
• Thousands and ten thousands continue this pattern, increasing exponentially with each step to the left.

In this exercise, recognizing the place value helps distinguish where to start borrowing. For instance, borrowing from the thousands for the hundreds when necessary by understanding each number's specific place value.

Mathematical Operations

Mathematical operations are the basic processes we perform with numbers, such as addition, subtraction, multiplication, and division. Subtraction, specifically, involves finding the difference between two numbers.

Here are the basic steps of subtraction:

• Align the numbers vertically by their place values — units with units, tens with tens, etc.
• Begin with the smallest place value (units) and move leftwards to larger place values.
• If the top number in any column is smaller than the bottom number, perform borrowing to make subtraction possible.
• Write down the result of each column subtraction, which collectively gives the final result.

In our subtraction exercise, following these fundamental subtraction operations allowed us to solve the problem efficiently as we subtracted each place value in order — resulting in the final difference.