Problem 31 - Algebra A Combined Function

Question

Perform the indicated operation. $$ \begin{array}{r} 654.9 \\ -\quad 56.67 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The result is 598.23.

Step 1: Align the Numbers

Write the numbers 654.9 and 56.67 one below the other. Ensure that the decimal points are vertically aligned, which means arranging them from right to left based on their decimal places.

Step 2: Append Zeros

Add a zero to the end of 654.9, making it 654.90. This helps in aligning the digits properly under each decimal place for subtraction.

Step 3: Subtract the Hundredths

Subtract the numbers starting from the rightmost digit (the hundredths place). For 0 - 7, borrow 1 from the tenths place (9 becomes 8, and 0 becomes 10). Thus, 10 - 7 = 3.

Step 4: Subtract the Tenths

Move to the tenths place: 8 (after borrowing) minus 6 equals 2.

Step 5: Subtract the Ones

In the ones place, we have 4 - 6. Borrow 1 from the tens place (5 becomes 4, and 4 becomes 14). 14 - 6 = 8.

Step 6: Subtract the Tens

In the tens place, subtract 4 from 5 (after borrowing), resulting in 9.

Step 7: Subtract the Hundreds

Finally, in the hundreds place, subtract 0 from 6, which gives 6.

Step 8: Write the Answer

Combine the digits from each step, ensuring to place the decimal point correctly, resulting in 598.23.

Key Concepts

Place Value

Understanding place value is fundamental to working with decimals, especially when performing operations like subtraction. Each digit in a number has a specific value depending on its position.
For example, in the number 654.9:

• 6 is in the hundreds place, which means it represents 600.
• 5 is in the tens place, representing 50.
• 4 is in the ones place, standing for 4.
• 9 is after the decimal point in the tenths place, corresponding to 0.9.

The same rule applies to other decimals like 56.67. Recognizing these positions helps you correctly align numbers and understand what values you are operating with during subtraction. Staying attentive to place values ensures accuracy in calculating results.

Borrowing in Subtraction

Borrowing is a crucial step in subtraction, especially when dealing with numbers where the top digit is smaller than the bottom digit in any column. When you need to subtract a bigger number from a smaller one, borrowing comes into play. For example:

• From the number 654.90 (prepared by aligning properly), you subtract 56.67.
• In the hundredths column, to subtract 7 from 0, you borrow 1 from the tenths column, turning the 0 into 10. After borrowing, the 9 in the tenths column becomes an 8.
• Similarly, borrowing occurs when subtracting 6 from 4 in the ones place. Borrow 1 from the tens place, turning the 4 into 14, then complete the subtraction.

By borrowing correctly, you ensure each subtraction is feasible, moving any extra value over to balance the operation.

Aligning Decimals

Aligning decimals is a vital first step in the subtraction of decimal numbers. This ensures that digits occupying the same place values are directly above one another, making the process straightforward and preventing errors. Here’s how to align decimals effectively:

• Write the numbers in a column format, ensuring the decimal points are one below the other, vertically aligned.
• Fill in any missing decimal places with zeroes to make all numbers have equal length past the decimal. This keeps the subtraction orderly.
• In our example, the numbers 654.9 and 56.67 become 654.90 and 56.67.

Proper alignment guarantees that each digit is subtracted from its corresponding place value, maintaining the integrity of the operation.

Arithmetic Operations

Understanding arithmetic operations is essential for performing decimal subtraction smoothly. Subtraction in decimals follows basic arithmetic rules, with additional awareness of the place value and alignment requirements.

Steps to Subtract Decimals:

• Align the numbers properly with decimal points in a vertical line.
• Add zeros where necessary to match the number of decimal places.
• Begin subtracting from the rightmost digit, moving left through each decimal place.
• Use borrowing wherever needed to handle larger bottom digits.

By observing these steps, you can accurately perform the operation and verify that all place values are correctly reduced. Following these arithmetic rules provides clarity and precision in dealing with decimals.