Problem 22 - Fundamentals of Mathematics

Question

For the following problems, perform the additions. If you can, check each sum with a calculator. $$ \begin{array}{r} 265 \\ +103 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum of 265 and 103 is 368.

Step 1: Set Up the Problem

Write the numbers one on top of the other with their digits aligned according to place value. Place the number 265 above the number 103, ensuring that the ones, tens, and hundreds places are lined up.

Step 2: Add the Ones Column

Add the digits in the ones column: 5 + 3 = 8. Write down the number 8 below the line in the ones place.

Step 3: Add the Tens Column

Move to the tens column and add the digits: 6 + 0 = 6. Write the number 6 below the line in the tens place.

Step 4: Add the Hundreds Column

Finally, add the digits in the hundreds column: 2 + 1 = 3. Write the number 3 below the line in the hundreds place.

Step 5: Combine the Results

The digits from the ones, tens, and hundreds columns give the sum of the two numbers. Write the complete sum below the line: 368.

Step 6: Verify with a Calculator

Use a calculator to double-check the addition: Enter 265 + 103 and ensure the result is 368, confirming that the manual calculation is correct.

Key Concepts

Place Value

Understanding place value is essential in addition. Each digit in a number has a position, known as its place value, which determines its actual value based on the column it occupies. For example, in the number 265, the '2' is in the hundreds place, meaning it represents 200, the '6' is in the tens place, representing 60, and the '5' is in the ones place, representing 5. Place value helps avoid mistakes during addition because it keeps numbers aligned according to their value.

• Hundreds Place: The leftmost digit is in the hundreds place, holding a value that is a multiple of 100.
• Tens Place: The middle digit, which represents a multiple of 10.
• Ones Place: The rightmost digit, representing a single unit.

Recognizing these places means you can add digits correctly across columns without mixing up their values.

Column Addition

Column addition is a methodical way of adding numbers, ensuring precision and clarity. Here, you write the numbers one below the other, aligning them by their place values. Start from the rightmost column and proceed left. In our example:

• Ones Column: Add digits in the ones place, 5 + 3 = 8.
• Tens Column: Move left and add the tens digits, 6 + 0 = 6.
• Hundreds Column: Add the hundreds digits, 2 + 1 = 3.

Don't forget to record each result directly below the line in their respective columns. Sum up all columns to get the total. This method not only simplifies the process but also minimizes errors during calculations. Column addition lets you clearly see and verify each step, making it optimal for handling larger numbers efficiently.

Verification with a Calculator

Verification with a calculator is an excellent way to confirm manual calculations. After solving the addition manually, simply input the original numbers into a calculator. This instant check is crucial for ensuring accuracy, especially in homework or tests where mistakes need to be minimized. To verify:

• Input: Type the numbers in (e.g., 265 + 103).
• Calculate: Observe the output, ensuring it matches your manual result (368).
• Confirm: Feel confident that your process and final answer are correct.

By verifying with a calculator, you not only bolster your accuracy but also strengthen your understanding of addition as any discrepancy can reveal areas for improvement. It builds a habit of checking work, which is beneficial in mathematical problem-solving.