Problem 53 - Prealgebra

Question

Perform the indicated operations. $$\begin{array}{r} 37 \\ +45 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The sum of 37 and 45 is 82.

Step 1: Align the Numbers

To add the numbers 37 and 45, start by aligning them vertically, with each digit in its respective column based on place value. \[\begin{array}{c} & 3 & 7 \+ & 4 & 5 \\hline\end{array}\]

Step 2: Add the Rightmost Column

Begin by adding the digits in the rightmost column. Add 7 and 5 together:\(7 + 5 = 12\).Since 12 is greater than 9, we write down 2 and carry over 1 to the next left column.

Step 3: Add the Leftmost Column

Now, add the digits in the leftmost column, including the carryover. The digits to add are 3, 4, and the carryover 1:\(3 + 4 + 1 = 8\).Write down 8 in the leftmost position.

Step 4: Write the Final Sum

Combine the results from the previous steps. The sum of 37 and 45 is 82:\[\begin{array}{c} & 3 & 7 \+ & 4 & 5 \\hline & 8 & 2\end{array}\]

Key Concepts

Place Value

Place value is a fundamental concept in mathematics that determines the value of a digit based on its position within a number. In a multi-digit number, each digit has a specific place value-ones, tens, hundreds, etc. This helps us understand the true value of each digit within the number.
For example, in the number 37:

• "3" is in the tens place, which means it represents 30 (or 3 tens).
• "7" is in the ones place, meaning it represents 7 (or 7 ones).

When adding numbers like 37 and 45, recognizing the place value of each digit helps ensure we combine values accurately and systematically. Ensuring that digits of the same place value are aligned vertically is key to performing addition correctly.

Carrying Over

Carrying over, often referred to as "regrouping," is an essential part of addition, especially when dealing with the rightmost column of numbers. When the sum of digits in a column exceeds 9, we need to carry the extra value over to the next column. This allows us to keep only a single digit in each place value column.
For instance, when adding 7 and 5:

• The sum is 12, which is greater than 9.
• Write down the "2" in the ones column.
• Carry over "1" to the tens column.

By carrying over correctly, we can ensure that each digit in our final sum is in the correct place value. This process helps us maintain the numerical integrity of the numbers we are adding.

Vertical Alignment

Vertical alignment in addition ensures that digits are lined up in a column format according to their place values. This alignment helps ensure accuracy and ease in the addition process by clearly delineating what digits should be added together.
Consider the alignment of 37 and 45:

• The digits "7" and "5" are vertically aligned. Both are in the ones column.
• The digits "3" and "4" are also vertically aligned. Both are in the tens column.

This organizational method is important because it minimizes errors and makes the addition of large numbers more manageable. Since each column is neatly lined up based on place values, we can easily perform addition starting from the rightmost column and moving left, applying carrying over as needed.