Problem 33 - Prealgebra

Question

Find each of the following differences. (Subtract.) $$765.432-234.567$$

Step-by-Step Solution

Answer

530.865

Step 1: Align the Numbers

Write the numbers vertically, ensuring the decimal points are aligned: \[ \begin{array}{c} \phantom{0}765.432 \ -234.567 \end{array} \]

Step 2: Subtract the Thousandths Column

Start from the rightmost column (thousandths). Subtract 7 from 2. Since 2 is smaller than 7, borrow 1 from the hundredths column, making it 12. Then, 12 - 7 = 5.

Step 3: Subtract the Hundredths Column

Now, move to the hundredths column. Since we borrowed 1, our number here is 2. Subtract 6 from 2. Borrow 1 from the tenths column. Then, 12 - 6 = 6.

Step 4: Subtract the Tenths Column

Go to the tenths column. After borrowing, we have 3. Subtract 5 from 3. Again, borrow 1 from the units column. Then, 13 - 5 = 8.

Step 5: Subtract the Units Column

Move to the units column. After borrowing, our number is 4. Subtract 4 from 4, which equals 0.

Step 6: Subtract the Tens Column

Proceed to the tens column. Subtract 3 from 6, which equals 3.

Step 7: Subtract the Hundreds Column

Finally, consider the hundreds column. Subtract 2 from 7, which equals 5.

Step 8: Write the Final Answer

Arrange the result from each computed column: \[ \begin{array}{c} \phantom{0}765.432 \ -234.567 \ \hline 530.865 \end{array} \] Hence, the difference is 530.865.

Key Concepts

Decimal Subtraction

Subtraction can take on added complexity when dealing with decimals, hence it's crucial to handle them correctly. Decimal subtraction involves two main components: ensuring the numbers are perfectly aligned and then performing the subtraction column by column, starting from the rightmost decimal point.

When working with decimals, it's essential to:

• Align the decimal points straight up and down. This ensures each digit is in its proper place value column.
• Subtract from right to left, beginning with the smallest column, or the farthest right decimal, moving through to the largest.

This method ensures that you preserve the value of each decimal place, maintaining accuracy in your subtraction.

Step by Step Math Solution

Breaking problems down into smaller, sequential steps is effective for solving decimal subtraction. Let’s walk through the process using the example of subtracting 234.567 from 765.432.

Begin by aligning the decimals; this will guide you through:

• Subtraction from the thousandths place and borrowing if necessary
• Subtracting through each subsequent column
• Borrowing from one column to maintain the operation when the top digits are smaller

These steps ensure clarity and help avoid errors. Writing your operations down onto paper or a calculator's display can help visualize each step too.

Borrowing in Subtraction

Borrowing is a technique used when you need to subtract a larger number from a smaller one in a column. This happens often in decimal subtraction, as seen in our example exercise.

Here's a quick run-down of borrowing:

• When subtracting two digits and the top digit is smaller, look to the left to borrow a '1' from the next immediate larger place value.
• Decrease the next left-hand digit by one to account for borrowing, raising the smaller place value by ten (or one order of magnitude higher depending on its tenths, hundredths position etc.).
• Apply this addition and perform the subtraction.

Understanding borrowing is critical in mastering decimal subtraction as it ensures you're performing your operations correctly, regardless of number size and sequence within subtraction.