Problem 11 - Fundamentals of Mathematics

Question

Find the sums and differences. $$ \begin{array}{r} 627 \\ +\quad 48 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum of 627 and 48 is 675.

Step 1: Understand the Problem

We need to find the sum of two numbers: 627 and 48. This means adding the two numbers together.

Step 2: Align the Numbers

Write the numbers one on top of the other, aligning them by the place value (ones, tens, hundreds). The number 627 goes on top, and 48 goes below it. Add a plus sign (+) before the number 48.

Step 3: Add the Ones Column

Add the numbers in the ones column: 7 (from 627) + 8 (from 48) equals 15. Write down 5 in the ones place of the sum and carry over 1 to the tens column.

Step 4: Add the Tens Column

Now, add the numbers in the tens column: 2 (from 627) + 4 (from 48) + 1 (carried over) equals 7. Write down 7 in the tens place of the sum.

Step 5: Add the Hundreds Column

Finally, add the numbers in the hundreds column: 6 (from 627) + 0 (as there is no digit in the hundreds place for 48) equals 6. Write down 6 in the hundreds place of the sum.

Step 6: Combine the Results

We now have all the digits of the sum: 675. Write down 675 as the final result of adding 627 and 48 together.

Key Concepts

Understanding Place Value

Place value is a fundamental concept in mathematics. It refers to the value of a digit based on its position in a number. Each position represents a different power of ten.

• The rightmost position is the ones place, representing numbers from 0-9.
• To the left of the ones place is the tens place, which groups in sets of ten, running from 10-90.
• Next is the hundreds place, which organizes in sets of one hundred. Each step leftward increases the grouping size by ten times.

In the addition problem with 627 and 48, understanding place values helps align numbers correctly. 627 has digits in the hundreds, tens, and ones places, while 48 has digits in the tens and ones places. Aligning them by place value ensures that you add the correct digits together.

The Art of Carrying Over

Carrying over is a key technique in addition. It occurs when the sum of digits in a column is greater than nine. This means you need to move the extra value to the next higher place value column.
Take the exercise where you add 627 and 48. In the ones column, 7 + 8 equals 15. Since 15 is greater than ten, you write down 5 and carry over the 1 (from 15) to the tens column. Carrying over ensures correct addition across all place values.

• In the tens column, you then add 2 from 627, 4 from 48, and the 1 carried over, totaling 7.
• The process of carrying over prevents mistakes and ensures precision in larger sums.

Mastering carrying over allows you to tackle complex addition problems with confidence.

Mastering Column Addition

Column addition is a straightforward technique, where numbers are aligned vertically by place value. This method simplifies addition by breaking it down into manageable parts. For the problem involving 627 and 48:

• Line up the digits according to their place values, stacking them into columns: ones, tens, and hundreds.
• You start from the rightmost column, add, and then work your way left, using carrying over as needed.
• This approach helps to neatly organize your work and lessen errors.

When completed, column addition provides a clear and organized way to achieve the final sum. It's a reliable method for solving addition problems of any size, ensuring you account properly for each digit.