Problem 32 - Fundamentals of Mathematics

Question

For the following problems, perform the additions. If you can, check each sum with a calculator. $$ \begin{array}{r} 75 \\ +\quad 6 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum is 81.

Step 1: Write Down the Numbers

Write down the numbers you want to add, aligning them by their rightmost digits. This means writing the numbers vertically in columns, with the unit digits aligned.

Step 2: Add the Units Place

Add the digits in the units column. Here, you are adding \(5 + 6\). The sum is \(11\). Write down \(1\) in the units place of the result and carry over \(1\) to the tens column.

Step 3: Add the Tens Place

Add the digits in the tens column along with any carry-over. Here, add \(7\) (from the number 75) and the carry-over \(1\). This gives you a sum of \(8\). Write this down in the tens place.

Step 4: Write the Final Result

Combine the digits from the sums we calculated. Place the \(1\) from the units place and the \(8\) from the tens place together to get \(81\).

Step 5: Verify with a Calculator

To ensure accuracy, re-check the addition using a calculator. Input the two numbers \(75\) and \(6\), perform the addition, and confirm that the result is \(81\).

Key Concepts

Vertical Alignment

When performing addition, it's crucial to align numbers vertically. Start by writing the numbers on top of each other, ensuring the rightmost digits line up perfectly. This alignment refers specifically to the unit digits forming a vertical column. For example, when adding 75 and 6, write 75 above 6 and line up the numbers by the rightmost digits. Doing so helps maintain proper order and allows for easier addition, particularly when dealing with large numbers. By keeping digits straight within their columns, you can avoid errors that arise from misalignment.

Carrying Over

Carrying over is a fundamental technique used when adding multi-digit numbers. It comes into play when the sum of digits in a particular column exceeds 9. In the example of adding 75 and 6, the unit digits (5 and 6) sum to 11. Here, you write 1 in the unit's column and "carry over" 1 to the next column, which is the tens place.

• Indicate the carried-over digit above the next column to add.
• In the tens place, add together the digits and any carried-over number. For our problem, add 7 (from 75) and the carried-over 1. This results in 8, which you write in the tens place.

This method ensures all columns are accurately summed and prevents mistakes in your final total.

Digit Addition

Digit addition is the process of adding each corresponding pair of digits from two numbers. Begin with the rightmost digits, which are in the unit's column. In our example, you add 5 and 6 to get 11.

• After dealing with any carry over, proceed left to the next column.
• Add the digits, adding any carry over from the previous column as well.

For 75 plus 6, after adding the unit digits and carrying over, move to the tens column. Here, you add 7 (tens digit of 75) and the carried-over 1, totaling 8. This way, digit addition progresses column-by-column until all digits are added.

Check with Calculator

Even after following all steps correctly, it's good practice to verify your result using a calculator. Recapping our previous steps: align the numbers, perform digit addition, and account for any carry over. But using a calculator provides additional peace of mind.
To check 75 + 6, key these numbers into the calculator. The display should confirm that the total is indeed 81.

• Re-checking helps find any unnoticed mistakes.
• It solidifies your understanding of the manual addition process.

Always make it a habit to validate your sums; it enhances your numerical accuracy and builds confidence in mathematics.