Problem 31 - Prealgebra

Question

Find each of the following differences. (Subtract.) $$8-0.327$$

Step-by-Step Solution

Answer

The difference is 7.673.

Step 1: Understand the Problem

We need to subtract 0.327 from 8. This means we are finding the difference between the number 8 and the decimal number 0.327.

Step 2: Align the Numbers Vertically

To subtract, align the numbers by their decimal points:\[\begin{array}{r} 8.000 \-0.327 \\hline\end{array}\]Notice that we've added three zeros to the right of the decimal point after the 8, so it becomes 8.000. This makes it easier to subtract decimals.

Step 3: Subtract the Hundredths Place

Subtract the hundredths place: 0 - 7. Since 0 is less than 7, we need to borrow. Borrow 1 from the tenths place. Now, it looks like:\[\begin{array}{r}7.9(10)0 \-\phantom{9}0.3(1)(1)\\hline\end{array}\]Now, 10 - 7 = 3. Write 3 under the line in the hundredths column.

Step 4: Subtract the Tenths Place

Now subtract the tenths place: 9 - 2. Since 9 is greater than 2, there is no need to borrow. The result is 7. Write this under the line in the tenths column.

Step 5: Subtract the Ones Place

Finally, subtract the hundredths place in the reduced number after borrowing: 7 - 0 = 7. Write 7 under the line in the whole number column.

Step 6: Record the Final Result

Combine the results from each place value to get the final answer, which is 7.673. Therefore, the difference between 8 and 0.327 is 7.673.

Key Concepts

Decimal Subtraction

When dealing with decimal subtraction, the first step is always to properly align the numbers so they can be subtracted accurately. This often means padding the numbers with zeros for better alignment. In our example, to subtract \(0.327\) from \(8\), we express \(8\) as \(8.000\).
This addition of zeros after the decimal doesn't change the value. It only extends the number to reflect the same number of decimal places as the number you're subtracting.

• Align decimal points to ensure that each digit falls into its respective place value.
• Add zeros where necessary to match the number of decimal places.

Decimal subtraction relies heavily on maintaining correct alignment throughout the calculation.

Borrowing in Subtraction

Borrowing in subtraction is a technique used when a digit in one column is smaller than the number you need to subtract from it. For example, when subtracting \(0.327\) from \(8.000\), you cannot directly subtract \(7\) from \(0\). Thus, borrowing is required.
When you borrow, you take 1 from the next higher digit column. Here, you borrow \(1\) from the tenths column and add \(10\) to the hundredths column.

• Borrowing allows you to perform subtraction even when the digit is smaller.
• Always start borrowing from the smallest place value that needs it.

Borrowing is a straightforward concept that can be practiced for mastery. It is an essential technique in decimal arithmetic.

Arithmetic Operations

Understanding arithmetic operations like decimal subtraction expands your basic math skills. Arithmetic operations include addition, subtraction, multiplication, and division, all of which can also be applied to decimals.
Each operation follows particular rules that streamline the process:

• Decimal addition and subtraction require aligning decimals.
• For subtraction, borrowing ensures each column has an adequate number of high-value digits to conduct subtraction accurately.
• When performing these operations, accuracy in maintaining each step determines the outcome.

Practice these arithmetic operations to improve your overall mathematical fluency, especially when working with decimals.