Problem 66 - Algebra 1: Concepts and Skills

Question

ADDING DECIMALS Add. $$ 2.008+1.10199 $$

Step-by-Step Solution

Answer

2.008 + 1.10199 equals 3.10999.

Step 1: Align Decimal Points

Firstly, make sure to align the decimal points. This can be done by writing one below the other like so: \[ \begin{aligned} &2.008 \ +&1.10199 \end{aligned} \]

Step 2: Add with Alignment

Next, begin adding from the rightmost digit, which is the decimal part. Just as with whole numbers, if a column creates a number 10 or greater, carry the 'extra' to the next column to the left. In this case, you'll notice that 2.008 doesn't align perfectly with 1.10199, treat unaligned numbers as if they are followed by zeros: \[ \begin{aligned} &2.00800 \ +&1.10199 \ \hline &3.10999 \end{aligned} \]

Step 3: Check Result

Lastly, check the result to make sure it looks reasonable. The result should not be smaller than either of the inputs, and it should have the decimal point in the correct place

Key Concepts

Decimal Alignment

When adding decimal numbers, it is crucial to make sure that the decimal points are aligned. This step ensures that each digit is correctly placed in its respective column, much like lining up numbers in a column method.
For instance, aligning the decimal points in our example of 2.008 and 1.10199 requires placing them one below the other using their decimal points as guides.

• Write 2.008 over 1.10199, aligning the decimal points.
• Add extra zeros to the shorter number to ensure that both numbers have the same number of decimal places.

Ensuring correct alignment prevents mathematical errors during calculation. Think of these zeros as placeholders that maintain the number's value during addition.

Adding Decimal Numbers

Once you have aligned your decimal points, you can proceed to add the numbers as you would with whole numbers. Begin the addition from the rightmost column of digits, which includes the decimal part.
In our example, after aligning, you would start adding from the smallest decimal place:

• Begin with the rightmost decimal place and move sequentially to the left.
• If the sum of a column is 10 or greater, carry over the extra to the next left column.

For example, adding 2.008 and 1.10199 involves aligning them with zeros added to ensure all decimal places are filled. The process is:
Align: \( 2.00800 + 1.10199 = 3.10999 \).
Lining up helps you to correctly add numbers of different lengths, ensuring accuracy and consistency in your results.

Decimal Point Placement

The placement of the decimal point in the result is an essential aspect of adding decimals. Once you complete your addition, the decimal point in the sum should be directly aligned with the decimal points from the numbers you added.
Here is how you verify that:

• Check if the decimal point is in a position directly below the decimal points in the numbers being added.
• Review each column's sum to confirm there haven’t been any misplaced carries which can shift the decimal point wrongly.

In our example, after adding 2.008 and 1.10199, the decimal point in the result \( 3.10999 \) aligns perfectly right where the original decimal points were. Proper decimal point placement ensures the integrity of the value and that no digits contribute to incorrect place value understanding.