Problem 39 - Prealgebra

Question

Add $$396+19.8$$

Step-by-Step Solution

Answer

The sum of 396 and 19.8 is 415.8.

Step 1: Align the Numbers

First, we need to write the numbers vertically, ensuring that the decimal points are aligned. We add zeros to the right of the whole number 396 to match the number of decimal places:\[ \begin{array}{c} 396.0 \+ 19.8\end{array} \]

Step 2: Add the Decimal Places

Start adding from the rightmost digit (tenths place). Add 0 (in 396.0) and 8 (in 19.8):\[ 0 + 8 = 8 \]

Step 3: Move to the Next Column (Units Place)

Add the next column of digits moving left, adding 6 (in 396) and 9 (in 19):\[ 6 + 9 = 15 \]Write 5 in the units place and carry over 1 to the tens place.

Step 4: Add the Tens Place

Now, add the numbers in the tens place along with the carry we have from the previous step:\[ 9 + 1 (carry) + 1 = 11 \]Write 1 in the tens place and carry over another 1 to the hundreds place.

Step 5: Add the Hundreds Place

Finally, add the numbers in the hundreds place:\[ 3 + 1 (carry) = 4 \]

Step 6: Write the Final Sum

By putting all the digits together, we get:\[ \boxed{415.8} \]

Key Concepts

Step-by-Step Solution

Tackling decimal addition can be simplified by approaching it methodically, using a step-by-step solution. This process allows you to break down the calculation into manageable parts, ensuring accuracy with each step.
When you add numbers, especially with decimals, ensure all steps are clear and incremental. Begin by laying out each problem step, as this removes confusion and builds confidence.
Start by writing both numbers in a vertical layout, paying special attention to aligning the decimal points. This positioning is crucial and will be discussed more thoroughly later. Continue by focusing on one operation at a time, moving through the sequence of addition from the rightmost digit to the leftmost.
Handle each column's sum individually, particularly if tackles involving carries occur, which need special attention to complete the solution correctly. Be patient, and proceed to verify each column's addition before moving on to the next. In the end, connect all the results together to get your final answer.

Aligning Decimal Points

One of the key aspects of adding decimals is ensuring that the decimal points are aligned vertically. This alignment ensures that each digit is placed in the correct column representing its respective place value.
Here's why this alignment is crucial: in decimal arithmetic, every number has a specific value based on its position relative to the decimal point. For example, in the number 19.8, the 9 is in the units place, and the 8 is in the tenths place.
Aligning decimal points in the vertical representation, like with the 396.0 and 19.8 example, helps keep track of these positions and avoids mixing numbers of different place values.

• Place the numbers one beneath the other, aligning the decimal points.
• Add zeros to match the number of decimal places when necessary, as was done with 396 becoming 396.0.

This technique allows for an easy and seamless addition process. By aligning correctly, the addition of the numbers from different precision levels becomes straightforward.

Carrying in Addition

Carrying is a fundamental concept when adding numbers that results in a sum greater than 9 in any column. This extra value needs to be 'carried over' to the next column to the left.
During the addition of 396 and 19.8, carrying occurs at two critical points:

• When adding 6 and 9 from the units place, the result is 15. The 5 stays in the units column, and the 1 is carried over to the tens column.
• Next, the tens column has a sum of 9 plus the carried 1, plus 1 from 19, totalling 11. Here, 1 remains in the tens column, and another 1 is carried to the hundreds column.

Carrying ensures each column results accurately reflect the true sum, maintaining the correct value as you progress left through the numbers.

Place Value Alignment

Understanding and utilizing place value alignment is critical in adding decimals correctly. Place value refers to the value a digit has based on its position within a number, such as hundreds, tens, or units, extending into decimal places like tenths or hundredths.
In our given problem, extending the number 396 to 396.0 aligns it with 19.8 by matching decimal places. These alignments tie directly into the value each digit holds. Aligning digits by their place value ensures that when you sum, for example, 6 and 9 (in the units place), they result in an accurate total.

• Units add units: secure the integrity of the basic number positions.
• Decimal places add decimal places: maintain precision by aligning tenths with tenths.

Each digit maintains its assigned numerical role, harvested from its precise placement in the arrangement of digits. This structured approach ensures that your final result is not only correct but accurately reflects the intended value.