Problem 24 - Prealgebra

Question

Find each of the following differences. (Subtract.) $$9.87-1.04$$

Step-by-Step Solution

Answer

The difference is 8.83.

Step 1: Align the Numbers

Write the numbers vertically, ensuring that the decimal points are aligned. This helps to keep track of place values.\( \begin{array}{c} 9.87 \ \- \ 1.04 \ \hline \end{array}\)

Step 2: Subtract the Hundredths

Start with the rightmost column, which is the hundredths place. Subtract the numbers in this column: \[7 - 4 = 3\]Write 3 under the line in the hundredths place.

Step 3: Subtract the Tenths

Move to the tenths place and subtract:\[8 - 0 = 8\]Write 8 under the line in the tenths place.

Step 4: Subtract the Ones

Finally, subtract the numbers in the ones column:\[9 - 1 = 8\]Write 8 under the line in the ones place.

Step 5: Write the Result

Combine the digits obtained from the subtraction to write the final result. Don’t forget to place the decimal point in the correct position:\[8.83\]

Key Concepts

Place Value

Understanding place value is essential for mastering decimal subtraction. Each digit in a decimal number represents a different power of ten, depending on its position relative to the decimal point. On the left of the decimal point, the digits represent whole numbers such as units, tens, hundreds, and so on. On the right side, the digits represent fractions of ten, such as tenths, hundredths, and thousandths.
For example, in the number 9.87:

• The '9' is in the ones place and represents 9 whole units.
• The '8' represents 8 tenths (0.8).
• The '7' represents 7 hundredths (0.07).

The alignment of decimal points is crucial as it ensures that each number is subtracted from its corresponding place value. Ignoring this can lead to incorrect calculations.

Step by Step Subtraction

Subtraction of decimals can be simplified by using a step by step approach. Begin by writing the decimals one below the other, aligning the decimal points. This sets you up for an organized and accurate subtraction process.
Here are the steps broken down:

• **Align the Numbers**: Place the numbers so that decimal points and the corresponding place values line up.
• **Start with the Rightmost Digit**: Start subtracting from the hundredths place and move left. This order ensures a systematic approach, similar to "borrowing" in regular subtraction.
• **Subtract and Write Each Result Below**: After subtracting each column, write the resulting digit below the line and to the correct decimal place.

Always remember to regroup, or borrow, from the next higher place if the upper digit is smaller than the lower digit. This keeps the subtraction process flowing logically and avoids errors.

Subtracting Decimals

Subtracting decimals is a straightforward extension of basic subtraction with the added step of managing decimal points and place values. Assure the precision of your answer by strictly maintaining alignment of the decimal points.
Consider each phase checking:

• **Correct Alignment**: Without ensuring alignment, place value missteps might occur, leading to incorrect results.
• **Carry Over**: Just like with whole numbers, sometimes borrow or "carry over" from the adjacent higher place value.
• **Fill in Missing Digits**: Use zeros to fill in any empty decimal places – this won't change the number's value but helps keep place values in check.

After completing the subtraction, double-check that the decimal point in your answer is directly beneath the decimal points of the numbers above. Keeping each task methodical ensures clarity and accuracy in your calculations.