Problem 9 - Prealgebra

Question

Find each of the following sums. (Add.) $$0.081+5+2.94$$

Step-by-Step Solution

Answer

The sum is 8.021.

Step 1: Align the Numbers

Before adding, align the numbers by their decimal points. This means you place the numbers in such a way that all decimal points are vertically aligned:\[\begin{array}{r}0.081 \5.000 \2.940 \\hline\end{array}\]

Step 2: Perform the Addition

Begin adding the numbers starting from the rightmost column (the thousandths place) and move leftwards. Remember to carry over if a sum of a column exceeds 9.- Thousandths column: 1 + 0 + 0 = 1- Hundredths column: 8 + 0 + 4 = 12 (write 2 and carry over 1 to the next column)- Tenths column: 0 + 0 + 9 + 1 (carried) = 10 (write 0 and carry over 1)- Units column: 0 + 5 + 2 + 1 (carried) = 8Place these results in a new row:\[\begin{array}{r}0.081 \5.000 \2.940 \\hline8.021\end{array}\]

Step 3: Confirm the Result

Ensure that all columns were added correctly by verifying each digit's placement from the previous step. Double-check the carrying and any potential miscalculations. The sum is confirmed as:\[ 8.021 \]

Key Concepts

Alignment of Decimals

When adding decimal numbers, it is crucial to ensure that each number is aligned according to its decimal point. This involves

• placing each number so that the decimal points are in a vertical line,
• lining up digits number according to their place value (units, tenths, hundredths, etc.).

While aligning, you may notice some spaces without digits. It's perfectly fine to fill these with zeros to make things clearer. For example, 5 becomes 5.000 when aligned with decimals like 0.081 and 2.94.

This alignment helps in correctly adding each column of digits according to their place values, which reduces the risk of making an error during addition. It's a helpful strategy to manage the complexities that come with varying lengths of decimal numbers.

Carrying Over in Addition

In addition involving decimals, just like whole numbers, there may be a need to "carry over" if a column sum exceeds 9. This is an essential step to ensure accuracy.

• Begin from the rightmost digit and move leftwards.
• If you add the digits of a column and the result is a two-digit number, the digit on the left needs to be carried over to the next column.

For example:

• In the hundredths column, if you sum 8 and 4 to get 12, you would write down 2 and carry 1 over to the tenths column.
• This technique continues across all other columns until all columns are summed accurately.

Carrying over ensures that every column maintains correct place value balance, ultimately guaranteeing a precise sum. It's crucial even when working with small decimals because misplacing a digit can lead to significant errors in the final sum.

Place Value in Decimals

Understanding place value is fundamental when working with decimals. Each digit in a decimal number has a specific place value which dictates its actual value in the number.

• The digit immediately after the decimal point represents the tenths place.
• The next digit represents the hundredths place.
• The pattern continues with the thousandths place and so on.

In our example, when adding numbers like 0.081, 5.000, and 2.940, you have to respect these place values while performing additions.

For instance, the smallest decimal

• in the thousandths place in 0.081 is added directly to the thousandths in 2.940.

Neglecting to adhere to place value rules can lead to misplaced digits, resulting in incorrect sums. In every addition problem involving decimals, this understanding acts as a foundation to apply other techniques like carrying over and alignment.