Problem 37 - Fundamentals of Mathematics

Question

For the following problems, perform the subtractions. You may chedk each difference with a calculator. $$ \begin{array}{r} 5,000,566 \\ -2,441,326 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The difference is 2,559,240.

Step 1: Arrange the Numbers

Write the numbers one below the other, aligning them by their rightmost digits. We have 5,000,566 at the top and 2,441,326 directly below it.

Step 2: Subtract Column by Column

Start subtracting from the rightmost column. If a digit in the minuend (top number) is smaller than the subtrahend (bottom number) in that column, borrow from the next left column.

Step 3: Subtract the Units Column

Subtract the rightmost digits: 6 - 6 = 0.

Step 4: Subtract the Tens Column

Subtract the tens digits. Since 6 is smaller than 2, borrow 1 from the hundreds place, making it 16. Now, subtract: 16 - 2 = 4.

Step 5: Subtract the Hundreds Column

After borrowing, instead of 5, you have 4 in the hundreds place. Subtract: 4 - 3 = 1.

Step 6: Subtract the Thousands Column

Subtract the thousands digits: 0 must borrow from the next place, making it 10 - 4 = 6 (after adjusting for borrowed 1).

Step 7: Subtract the Ten Thousands Column

In the ten thousands place, 9 is left after borrowing one. Subtract: 9 - 4 = 5.

Step 8: Subtract the Hundred Thousands Column

Subtract: 9 - 2 = 7.

Step 9: Subtract the Millions Column

Subtract the remaining 5: 5 - 2 = 2.

Step 10: Write Down the Difference

Write down the result from the top to the bottom of calculated losses: 2,559,240.

Key Concepts

Borrow Operation

When performing subtraction, you might encounter a situation where you need to subtract a larger digit from a smaller one. This is where the borrow operation comes into play. It allows you to "borrow" value from a higher place value in the number.

For example, if you need to subtract 6 from 2 in the tens column, you can't subtract directly since 2 is smaller than 6. Instead, you borrow 1 from the hundreds place, turning the 2 into 12. Now, you can perform the subtraction:

• Before borrowing: 2 - 6
• After borrowing: 12 - 6 = 6

The borrow operation ensures that subtraction can proceed smoothly, even when digits are smaller in the minuend than in the subtrahend.

Place Value

Place value is a fundamental concept in arithmetic where the position of a digit determines its value. Each digit in a number has a place, such as units, tens, hundreds, and so on, which contributes to the overall value depending on its position.

For example, in the number 5,000,566, each digit has a specific place value:

• 6 in the units place
• 6 in the tens place
• 5 in the hundreds place
• 0 in the thousands place
• 0 in the ten-thousands place
• 5 in the millions place

Understanding place value is crucial when performing operations like subtraction because it tells us how much each digit is worth. This understanding is particularly important when borrowing, as you must borrow from the correct place value.

Column Subtraction

Column subtraction is a method where you subtract numbers by working through each column of digits, starting from the rightmost (units column) and moving left.

To perform column subtraction:

• Align the numbers so that each digit is in the correct column based on place value.
• If the top digit is smaller than the bottom digit, use borrowing from the next column to the left.
• Subtract each column, recording the result below the line.

This method is systematic, careful, and particularly useful for large numbers, like 5,000,566 minus 2,441,326. It ensures that each digit is handled according to its place value and makes borrowing straightforward.

Minuend and Subtrahend

In subtraction, the minuend and subtrahend are names for the numbers you're working with.

• The minuend is the number you are subtracting from, usually the larger number, placed on top.
• The subtrahend is the number you subtract, placed underneath the minuend.

For example: when subtracting 2,441,326 from 5,000,566, 5,000,566 is the minuend, and 2,441,326 is the subtrahend.

Recognizing these terms is beneficial not only for solving subtraction problems but also for understanding mathematical vocabulary as a whole. Knowing what's being subtracted from what can also guide us in strategies like borrowing.