Problem 50 - Fundamentals of Mathematics

Question

For the following problems, perform the subtractions. You may chedk each difference with a calculator. $$ \begin{array}{r} 64,000,002 \\ -\quad 856,743 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

63,143,259

Step 1: Understanding the Problem

We need to perform a subtraction between two numbers: 64,000,002 and 856,743. The larger number is the 'minuend' (64,000,002) and the smaller number is the 'subtrahend' (856,743).

Step 2: Aligning the Numbers

Write the numbers in a vertical format, making sure to align them according to their place value (ones, tens, hundreds, etc.). This facilitates accurate subtraction: \(\begin{array}{r} 64,000,002 \ - \ \ \quad 856,743 \ \hline \end{array}\).

Step 3: Subtract Units Position First

Subtract the digits in the units column (far right). Since 2 is smaller than 3, we need to borrow 1 from the tens column. This makes it "12 minus 3" giving a result of 9.

Step 4: Borrow and Subtract Tens Position

The tens column now becomes '0' (after borrowing 1). To subtract 4 from 0, we need to borrow from the hundreds column, changing the zero to '10'. Now perform '10 - 4', giving 6.

Step 5: Continue Borrowing and Subtraction for Hundreds

The hundreds column now shows 0 again. Borrow from the thousands (which becomes 9) making this a '10'. Subtract '7' from '10', resulting in 3.

Step 6: Subtract Thousands

The thousands column is now 9 after borrowing. Perform '9 - 6', resulting in 3.

Step 7: Borrow for Ten Thousand Column

For tens of thousands, since it is 0 after borrowing again, we borrow from the hundred thousands (changing it to 9) and perform '10 - 5', resulting in 5.

Step 8: Subtract Hundred Thousands

The hundred thousands column now shows 9. Subtract 8, giving 1.

Step 9: Write Down Result for Million Place

Since there's no more borrowing needed, subtract as is: '3 - 0' resulting in 3.

Step 10: Combine Results

After subtracting all columns, combine the numbers from each column to get the final result: 63,143,259.

Step 11: Verification with Calculator

Check the final result by using a calculator to subtract 856,743 from 64,000,002 to ensure it matches the manual calculation.

Key Concepts

Borrowing in Subtraction

Borrowing is a crucial concept in subtraction, especially when dealing with larger numbers. When the digit in the minuend (the number being subtracted from) is smaller than the digit in the subtrahend (the number subtracted), borrowing becomes necessary. Here's how it works:

• Look at the units place first. If you cannot subtract, you must "borrow" from the next left column. This means decreasing the digit in the next left column by one, increasing the current column's digit by ten.
• For example, in 64,000,002 minus 856,743, the unit place has a 2 and a 3. Since 2 is less than 3, borrow 1 from the tens column, making it 12 minus 3, resulting in 9.
• This process might set off a chain of borrowing across multiple columns if there are consecutive small digits.

Understanding borrowing helps ensure you correctly perform every subtraction step, no matter how many digits are involved.

Place Value Alignment

Aligning numbers according to their place value is essential for accurate subtraction. It ensures that each digit is correctly placed under its equivalent, preventing errors.

• The place value system includes units, tens, hundreds, thousands, and so on, moving leftwards.
• In the example subtraction problem, write 64,000,002 over 856,743, aligning from the rightmost unit (1s) to the leftmost (millions) position.
• This alignment allows for clear borrowing and subtraction, as each digit pairs correctly with its direct counterpart.

Correct alignment is like setting up a roadmap that guides every borrowing action and subtraction, making the process straightforward and consistent.

Arithmetic Verification

Verifying your subtraction arithmetic can save you from mistakes. This process confirms the accuracy of your work, ensuring the answer is reliable.

• Re-calculate using the reverse operation, which in the case of subtraction means performing an addition. Add your subtraction result to the subtrahend (the number you subtracted).
• For instance, after calculating 64,000,002 - 856,743 = 63,143,259, add 63,143,259 and 856,743.
• If your addition brings you back to the original minuend (64,000,002), your subtraction is verified as correct.

Using a calculator can also serve as an effective tool for verification, offering peace of mind that the calculated difference is indeed accurate.