Problem 55 - Fundamentals of Mathematics

Question

For the following problems, perform the additions and round to the nearest hundred. $$ \begin{array}{r} 928,725 \\ 15,685 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

944,400

Step 1: Perform the Addition

First, add the two numbers together: - Start by adding the units place: 5 + 5 = 10, write down 0 and carry over 1. - Next, add the tens place: 2 + 8 = 10 + 1 (carried over) = 11, write down 1 and carry over 1. - Then, add the hundreds place: 7 + 6 = 13 + 1 (carried over) = 14, write down 4 and carry over 1. - Now, add the thousands place: 8 + 5 = 13 + 1 (carried over) = 14, write down 4 and carry over 1. - Next, add the ten-thousands place: 2 + 1 = 3 + 1 (carried over) = 4, write down 4. - Finally, add the hundred-thousands place: 9 + 0 = 9. The result of the addition is 944,410.

Step 2: Round to the Nearest Hundred

To round 944,410 to the nearest hundred, identify the hundreds digit, which is 4 (in the 410 part). Look at the digit to the right, which is 1. Since 1 is less than 5, we round down. Thus, 944,410 rounded to the nearest hundred is 944,400.

Key Concepts

Rounding Numbers

Rounding numbers is a way to simplify them, making calculations easier and more intuitive. When you round a number, you approximate it to a certain decimal place. Common rounding places include ten, hundred, and thousand.
To round to the nearest hundred, you must first identify the digit in the hundreds place. In 944,410, the hundreds digit is 4. Then, look at the digit immediately to the right, which is the tens digit. Here, it is 1.

• If the tens digit is 5 or above, you round up. This means you increase the hundreds digit by one. For example, if it were 5, the number would be rounded to 944,500.
• If the tens digit is less than 5, as in our example (1), you round down. The hundreds digit stays the same, resulting in the number 944,400.

Rounding helps in estimating and ensuring numbers are manageable for mental calculations. Whether you're budgeting or adjusting measurements, being comfortable with rounding enhances numerical fluency.

Place Value

Understanding place value is crucial in mathematics, particularly in operations like addition and subtraction. Place value refers to the value of each digit in a number based on its position. For numbers like 944,410:

• The farthest right digit, 0, is in the units place. It represents 0.
• The digit to the left, 1, is in the tens place, representing 10.
• The digit 4 to the left of that is in the hundreds place, representing 400.
• Another 4 in the thousands place, representing 4,000.
• 4 in the ten thousands place, representing 40,000.
• 9 in the hundred thousands place, representing 900,000.

Knowing place value helps in breaking down numbers and performing operations accurately. In our example, keeping track of each digit's value allows for precise addition before rounding the sum.

Carrying in Addition

Carrying in addition is a technique used when the sum of a column exceeds 9. It involves transferring part of this sum to the next column on the left. Let’s explore this with an example:
When adding 928,725 and 15,685:

• In the units place, 5 + 5 equals 10. Write down 0 and carry over the 1 to the tens place.
• In the tens place, 2 + 8 equals 10 plus the carried 1, equaling 11. Write down 1 and carry over 1 to the hundreds place.
• In the hundreds place, 7 + 6 equals 13 plus the carried 1, equaling 14. Write down 4 and carry over 1.
• This pattern continues through the thousands and ten-thousands places, where each sum requires writing the last digit and carrying the remainder over.
• In the hundred-thousands place, no carrying is needed as the digit's total (9+0) remains under 10.

This method ensures accuracy and efficiency in addition. It’s especially useful for performing calculations by hand, maintaining precision during complex operations.