Problem 26 - Algebra A Combined Function

Question

Perform the indicated operation. $$ \begin{array}{r} 65.0028 \\ 5.0903 \\ +6.9 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The result of the addition is 76.9931.

Step 1: Align the Numbers

Align the numbers based on their place values. This includes lining up the decimal points so you can easily add each column:\[\begin{array}{r} 65.0028 \ 5.0903 \ + 6.9000 \\hline\end{array}\]

Step 2: Add Column by Column

Start adding from the rightmost column (thousandths) to leftmost (tens). Make sure to carry over if a column adds up to more than 9: 1. Thousandths: 8 + 3 + 0 = 11 (write 1 and carry over 1) 2. Hundredths: 2 + 0 + 0 + 1 = 3 3. Tenths: 0 + 9 + 9 = 18 (write 8 and carry over 1) 4. Units: 5 + 5 + 6 + 1 = 17 (write 7 and carry over 1) 5. Tens: 6 + 1 = 7

Step 3: Write the Result

Combine all the sums from each column to write the final result:\[\begin{array}{r} 65.0028 \ 5.0903 \ + 6.9000 \\hline 76.9931 \\end{array}\]The result of the operation is 76.9931.

Key Concepts

Place Value Alignment

When performing decimal addition, it is crucial to align numbers based on their place values. This ensures that we add numbers that represent the same place value together, and it prevents errors in calculation. In the given example, aligning means lining up the decimal points of 65.0028, 5.0903, and 6.9. This allows you to see each digit corresponds correctly:

• The digits to the left of the decimal point represent whole units and tens.
• The digits to the right of the decimal point represent tenths, hundredths, and so on.

By aligning decimal points, operations on decimals become straightforward and error-free, allowing an accurate addition process.

Carrying Over in Addition

In arithmetic addition, especially with larger numbers or multiple decimals, carrying over is a critical step. Carrying over occurs when the sum of digits in any place value column exceeds 9. When this happens, you write down the unit digit of that sum and carry the tens digit over to the next column on the left.
For instance, in the problem provided, while adding the tenths: 0 + 9 + 9, we get 18.

• You write down the 8 in the tenths place.
• Carry the 1 (representing 10) over to the next column, the unit's place.

This carry-over ensures the correct total by accounting for any extra value that exceeds a single digit in any column.

Columnar Addition

Columnar addition is a method of adding numbers in a column format, from the rightmost column to the leftmost. This technique is particularly useful for adding decimals or large numbers since it breaks the task down into manageable parts.
In the given example, you begin at the rightmost digit (thousandths) and move leftwards across each aligned column:

• Start with the smallest place value (thousandths here) and add all respective digits down the column.
• Consider any carried-over numbers from the previous column before writing down the result.
• Proceed to the next column, moving towards the left, adding accordingly and carrying over if needed.

The systematic approach of columnar addition ensures precise arithmetic operations, as each digit is handled one column at a time.

Arithmetic Operations

Arithmetic operations such as addition, subtraction, multiplication, and division form the basis of all calculations. In the context of decimal addition, it is important to approach these operations methodically.
Addition, being one of the fundamental arithmetic operations, involves combining two or more numbers to generate a sum. Using a structured addition approach prevents mistakes, especially with decimals:

• Begin by understanding the values and align them as per place values.
• Perform the addition starting from the rightmost side, taking into account any carry-overs.
• Verify the sum by checking each column division for accuracy.

These steps, when carefully followed, lead to accurate results, which is key when performing arithmetic operations with decimals.