Problem 18 - Prealgebra

Question

Find each of the following sums. (Add.) $$\begin{array}{c}57.4698 \\\9.89 \\\32.032 \\\572.0079 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The sum is 671.3997.

Step 1: Align the Numbers

Start by writing the numbers vertically, aligning the decimal points to ensure the corresponding place values are in the correct columns:\[\begin{array}{c} 57.4698 \ 9.8900 \ 32.0320 \ 572.0079 \\hline\end{array}\]

Step 2: Add the Thousandths Column

Add the digit from each number in the thousandths place (rightmost column):\[8 + 0 + 0 + 9 = 17\]Write 7 in the thousandths place of the answer and carry over 1 to the hundredths column.

Step 3: Add the Hundredths Column

Next, add the digits in the hundredths place, including the carried over 1:\[9 + 9 + 3 + 0 + 1 = 22\]Write 2 in the hundredths place and carry over 2 to the tenths column.

Step 4: Add the Tenths Column

Add the tenths place digits, including the carried over 2:\[4 + 8 + 0 + 0 + 2 = 14\]Write 4 in the tenths place and carry over 1 to the units column.

Step 5: Add the Units Column

Add the digits in the units place with the carried over 1:\[7 + 9 + 2 + 7 + 1 = 26\]Write 6 and carry over 2 to the tens column.

Step 6: Add the Tens Column

Add the digits in the tens column, including the carried over 2:\[5 + 0 + 3 + 2 + 2 = 12\]Write 2 in the tens column and carry over 1 to the hundreds column.

Step 7: Add the Hundreds Column

Finally, add the digits in the hundreds column (only 572 has digits here) and the carried over 1:\[5 + 1 = 6\]Write 6 in the hundreds column.

Step 8: Write the Final Sum

Combine all the results from each column to get the final answer:\[671.3997\]

Key Concepts

Place Value Alignment

When adding decimals, aligning the place values is essential to ensure accurate results. This means you need to line up the decimal points vertically, so that each column represents the same place value.
This includes units, tenths, hundredths, etc. Aligning these ensures each number is correctly added to its corresponding part of the other numbers.
For example, by aligning 57.4698 and 9.89, the decimals line up, allowing you to add zeros to 9.89, making it 9.8900.

• This makes sure that digits in similar positions are added correctly.
• Zeros are placeholder digits that help maintain clarity, especially for numbers with fewer decimal places.

Accurate alignment prevents errors and ensures each place value is added with precision.

Carrying Over

The process of carrying over happens whenever you add digits and the sum exceeds 9. Just like in standard arithmetic, this rule applies to decimal addition as well.
When the total of a column exceeds the digit 9, you write down the last digit of the sum in the column and carry over the remaining value to the next higher column.
Consider adding thousandths:

• If 8 + 0 + 0 + 9 = 17, you write 7 in the thousandths and carry over the 1 to the hundredths.
• This process continues through all columns.

The concept of carrying ensures each column contributes correctly to the total sum.

Column Addition

Column addition is a structured way to add multi-digit numbers. By focusing on one column at a time, you manage the sum more systematically.
This method involves adding all numbers in one vertical column starting from the rightmost column and moving left.
In this technique:

• You start from the smallest values, such as the thousandths, then progress to larger values.
• It makes handling carry-overs efficient, as each level is handled sequentially from right to left.

Column by column, this ensures that numbers are not mixed, allowing for a smooth and error-free addition process.

Decimal Arithmetic

Decimal arithmetic involves operations, like addition, but with decimal numbers which include values less than one. When adding decimals:

• Keep decimal points aligned to ensure correct place value alignment.
• Use placeholders (zeroes) to keep columns uniform when necessary.

This arithmetic is slightly different from whole numbers due to fractions of a unit (e.g., 0.1 or 0.01).
Understanding this helps in adding decimals accurately by reminding us to treat them like regular numbers with fractional parts.
Therefore, focusing on decimals highlights the significance of accuracy in alignment and handling each value's precise worth.