Problem 68 - Algebra 1: Concepts and Skills

Question

Find the sum. $$0.009+9$$

Step-by-Step Solution

Answer

The sum of 0.009 and 9 is 9.009.

Step 1: Set up the Addition with Decimal Alignment

Align the numbers so that the decimal points are vertically in line with each other. This would look like this: \[ \begin{align*} &9\ &+\ &0.009\ \end{align*} \]Here, we write 9 as 9.000 because it helps in aligning the decimals correctly.

Step 2: Perform the Addition

Add the numbers column by column, starting from the right (the thousandths place), just like how normal addition is done. The sum is then: \[ \begin{align*} &9.000\ &+\ &0.009\ &-----\ &9.009\ \end{align*} \]

Key Concepts

Aligning Decimals

When adding decimal numbers, it's crucial to align the decimal points vertically. This ensures that each digit is in the correct place value position relative to others. For example, in the problem of adding \(0.009\) and \(9\), we first write the number \(9\) as \(9.000\).
This adjustment doesn't change its value but helps in correctly aligning the decimal points. By doing so, you create a tidy column of digits, much like a typical addition problem, which simplifies the process.

• Ensure the decimal points line up vertically.
• Add zeros to whole numbers to fill any gaps, maintaining equal decimal places.

By following these steps, you make sure each digit falls under the correct column for easier addition.

Place Value

Place value is a fundamental concept in mathematics that involves understanding the value of each digit in a number based on its position. This is particularly important when working with decimals. Each position to the right of the decimal point represents a fraction of ten, such as tenths, hundredths, or thousandths.
For the addition of \(9\) (written as \(9.000\)) and \(0.009\), we observe that:

• The digit in the tenths place for both numbers is \(0\).
• The digit in the hundredths place is also \(0\).
• The digit in the thousandths place is \(0.009\), which contains the \(9\).

Recognizing and properly using place value ensures that digits are correctly aligned during calculations, resulting in accurate sums.

Basic Addition

Basic addition is all about combining numbers to get a sum. Once decimal numbers are aligned based on their place value, the addition process is straightforward.
Begin adding from the rightmost column, which is the smallest place value, typically involving the smallest numbers. For our example, we start with the thousandths place:

• Add \(0\) in the thousandths of \(9.000\) with \(9\) in \(0.009\), resulting in a digit of \(9\) in the thousandths.
• Move to the hundredths and tenths, where no additional values need to add up, just carry over zeros where applicable.
• Finally, add \(9\) with \(0\) in the units to keep it as \(9\).

This results in the final sum of \(9.009\), demonstrating that with correctly aligned decimals and proper understanding of place value, basic addition is a breeze.