Adding and Subtracting Decimals

Key Concepts

Adding and subtracting decimals involves understanding the following key concepts:

• Aligning Decimals by Place Value
• Adding Decimals
• Subtracting Decimals
• Carrying and Borrowing

Aligning Decimals by Place Value

When adding or subtracting decimals, it is crucial to align the numbers by their place value. This means lining up the decimal points so that each digit in the same place value position is directly above or below the corresponding digit in the other number.

Example: To add 12.34 and 5.678, align the numbers as follows:

      12.34
    +  5.678
    

Adding Decimals

To add decimals, follow these steps:

1. Align the decimal points.
2. Add the digits in each place value column, starting from the rightmost column (hundredths place).
3. If the sum of any column exceeds 9, carry over the extra value to the next higher place value column.
4. Place the decimal point in the result directly below the decimal points of the addends.

Example: Add 12.34 and 5.678.

      12.340
    +  5.678
    --------
      18.018
    

Subtracting Decimals

To subtract decimals, follow these steps:

1. Align the decimal points.
2. Subtract the digits in each place value column, starting from the rightmost column (hundredths place).
3. If the digit in the top number is smaller than the digit in the bottom number, borrow 1 from the next higher place value column.
4. Place the decimal point in the result directly below the decimal points of the numbers being subtracted.

Example: Subtract 5.678 from 12.34.

      12.340
    -  5.678
    --------
       6.662
    

Carrying and Borrowing

Carrying and borrowing are essential techniques used in addition and subtraction, respectively. Carrying involves adding an extra value to the next higher place value column when the sum of a column exceeds 9. Borrowing involves taking 1 from the next higher place value column when the digit in the top number is smaller than the digit in the bottom number.

Example: In the addition of 12.34 and 5.678, carrying occurs when adding the tenths place (3 + 6 = 9).

Example: In the subtraction of 5.678 from 12.34, borrowing occurs when subtracting the thousandths place (0 - 8).

Examples and Analogies

Imagine you are stacking blocks of different heights to build a tower. Each block represents a digit in a decimal number. To add the heights of the blocks, you stack them on top of each other, aligning them by their base (decimal point). If a stack exceeds a certain height, you carry over the extra height to the next stack.

Similarly, when subtracting the heights of blocks, you may need to borrow height from the next higher stack if the top block is shorter than the bottom block.

Practical Application

Understanding how to add and subtract decimals is essential for everyday tasks such as calculating prices, measuring distances, and solving mathematical problems. By mastering these operations, you can perform accurate calculations and make informed decisions based on numerical data.