Problem 35 - Fundamentals of Mathematics

Question

For the following problems, perform the subtractions. You may chedk each difference with a calculator. $$ \begin{array}{r} 800 \\ -650 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The result is 150.

Step 1: Align the Numbers

Ensure that the numbers 800 and 650 are aligned vertically with the hundreds, tens, and units places in their correct columns.

Step 2: Subtract the Units Column

Begin subtracting from the smallest place value, which is the units column. Here, both numbers have zeros in the units place, so the subtraction is simple: \(0 - 0 = 0\).

Step 3: Subtract the Tens Column

Move to the tens column. Subtract 5 from 0. Since we cannot subtract 5 from 0 directly, we need to borrow 1 from the hundreds column, making the tens column 10. So, \(10 - 5 = 5\). Adjust the hundreds column accordingly, reducing it from 8 to 7.

Step 4: Subtract the Hundreds Column

Now subtract the hundreds. After borrowing, we have 7 in the hundreds place for the top number. Subtract 6 from 7: \(7 - 6 = 1\).

Step 5: Verify with Caculator

Use a calculator to verify the result of 150 by entering the subtraction \(800 - 650\). This should confirm the manual computation.

Key Concepts

Borrowing in Subtraction

Borrowing in subtraction is a technique used when digits in a particular column cannot be subtracted directly due to the bottom number being larger. It involves borrowing '1' from the next highest place value to make subtraction possible.
For example, in the subtraction problem of 800 minus 650, when attempting to subtract the tens column, you encounter 0 minus 5. Without borrowing, this calculation can't be completed because you can't take 5 away from 0.

• You look to the hundreds column, which has an 8.
• Borrow 1 from this 8, turning it into a 7.
• Transform the tens column's 0 into a 10 by adding the borrowed 1.

Now you can solve the tens column subtraction easily with 10 minus 5, which equals 5. Borrowing can seem tricky at first, but with practice, it becomes an intuitive part of the manual subtraction process.

Place Value Alignment

Place value alignment is crucial when performing subtraction manually. It ensures that digits of equal value—such as units, tens, and hundreds—are subtracted directly from one another.
In the example of subtracting 650 from 800, you must align the numbers vertically before beginning your calculations:

• Place the 800 above the 650.
• Align the hundreds, tens, and units columns clearly.

With proper alignment:

• Units (0 and 0) can be subtracted directly.
• Tens come next (after borrowing if necessary).
• Finally, handle the hundreds.

This alignment helps prevent mistakes and simplifies the subtraction task, especially with larger numbers.

Manual Subtraction Verification

After performing a manual subtraction, it’s a good idea to verify your result to ensure accuracy. One straightforward way is by using a calculator.
For example, after finding that 800 minus 650 equals 150, input these values into a calculator to double-check your work.

• Enter 800, then the subtraction sign (-), and 650.
• Press the equals sign (=) to get the result.

If the calculator matches the manual result of 150, your calculations are correct. Verification not only confirms the answer but also boosts confidence in numerical processing skills. Developing this habit helps catch errors that may happen during manual computations.