Problem 30 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

598.23

Step 1: Align the Decimal Points

Before subtracting, it's important to write the numbers such that their decimal points are aligned vertically. This ensures each digit is in its correct place value. Write 654.9 and 56.67 as follows: ``` 654.90 - 56.67 ``` Now, we can perform the subtraction operation.

Step 2: Subtract from Right to Left

Subtract the numbers column by column starting from the rightmost side. Begin with the hundredths place: 1. **Hundredths Place**: Subtract 7 from 0. Borrow 1 from the tenths place, making it 10. 10 - 7 = 3. 2. **Tenths Place**: After borrowing, subtract 6 from 8 (original 9 was reduced by 1 when borrowing), which leaves 3. 3. **Units Place**: Subtract 6 from 4. Borrow 1 from the tens place to make it 14. 14 - 6 = 8. 4. **Tens Place**: After borrowing, subtract 5 from 4 (original 5 was reduced by 1 when borrowing), which leaves 9. 5. **Hundreds Place**: Substract 5 from 6. 6 - 5 = 1.

Step 3: Write Down the Result

After performing the subtraction for each column: ``` 654.90 - 56.67 -------- 598.23 ``` The final result of the subtraction operation is 598.23.

Key Concepts

Place Value Alignment

For a successful decimal subtraction, aligning the place values is crucial. This ensures each digit is placed in its proper column, which preserves the value of the numbers. When subtracting numbers, especially with decimal points, each place value must line up vertically:

• Hundreds
• Tens
• Units/Ones
• Tenths
• Hundredths

In our exercise, 654.9 should be rewritten as 654.90. This ensures the decimal points are aligned, and every digit is in the right place. With this clear layout, confusion is minimized and the subtraction becomes straightforward.

Borrowing in Subtraction

Borrowing is a technique used when the top digit is smaller than the bottom digit in a subtraction column. It allows us to "carry" a value over to solve the issue.
When we borrow:

• "Borrow" means taking one unit from the left column.
• Convert this borrowed unit into a larger number (usually ten) in the current column.
• This helps in subtracting comfortably without negatives.

In the given problem, borrowing occurs in the hundredths, units, and tenths places.
For instance, since 0 is smaller than 7 in the hundredths, we borrow 1 from the tenths place. This changes 0 to 10, making subtraction feasible: 10 - 7 = 3.

Step-by-Step Solution

To tackle decimal subtraction, breaking it down into steps makes it manageable and clear:
1. **Align the Numbers**: Stack the numbers with their decimal points aligned to establish clear place values. 2. **Begin Subtraction on the Right**: Start subtracting from the rightmost column, moving leftwards. 3. **Handle Borrowing**: If the top digit is smaller, borrow from the next left column.
Following these steps, you ensure each digit is correctly subtracted. In the example solved, after aligning, we perform column subtraction from right to left with necessary borrowing, as detailed earlier.
This methodical approach ensures no steps are missed and helps in verifying the accuracy of your solution.

Hundredths Place

The hundredths place is the second digit to the right of the decimal, representing one-hundredth of a whole.
In our subtraction problem:

• We have digits 0 and 7 in the hundredths place.
• Initially, 0 cannot subtract 7, requiring a borrow from the tenths column.
• After borrowing, 0 becomes 10, and the subtraction proceeds: 10 - 7 = 3.

Understanding this place value is important when working with precision. Each hundredth represents a tiny fraction of the whole, yet crucial in getting the exact answer in decimal operations. Proper handling during subtraction, like borrowing, maintains numeric accuracy.