Problem 63 - Fundamentals of Mathematics

Question

For the following problems, perform the additions and round to the nearest hundred. $$ \begin{array}{r} 9,573 \\ 101,279 \\ 122,581 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum is 233,400 when rounded to the nearest hundred.

Step 1: Align the Numbers for Addition

Align the numbers vertically so each digit is in the correct place value column:\[ \begin{array}{rrr} & 9,573 \+ & 101,279 \+ & 122,581 \\hline\end{array} \]

Step 2: Perform the Addition

Add the numbers starting from the rightmost digit to each left place value:\[ \begin{array}{rr} & 9,573 \+ & 101,279 \+ & 122,581 \\hline & 233,433 \\end{array} \]The total sum of these numbers is 233,433.

Step 3: Round the Sum to the Nearest Hundred

To round the number to the nearest hundred, look at the tens digit. If it is 5 or greater, round up. If it is less than 5, round down. In 233,433, the tens digit is 3, which is less than 5, so we round down:\[ 233,433 \approx 233,400 \]

Key Concepts

Addition

Addition is one of the most basic arithmetic operations. It involves finding the total or sum by combining two or more numbers. In our exercise, we sum three numbers: 9,573, 101,279, and 122,581. To do this efficiently, it's crucial to align the numbers according to their place values, which means lining up the digits according to their respective columns: units, tens, hundreds, thousands, and so on.

• Start from the rightmost digit, which is the units place, and move to the left.
• Add digits vertically, carrying over any extra value to the next left column if the sum of a digit exceeds 9.

Once you perform these additions correctly, you aggregate all partial sums to get the total sum. For these specific numbers, the sum is 233,433.

Place Value

The concept of place value is foundational in arithmetic, including addition and rounding. It refers to the value of the digit depending on its position in the number. Understanding this is essential when performing operations like addition, as it ensures numbers are properly aligned. Each position in a number has a value ten times that of the position to its right. For example:

• The rightmost digit is the 'units' or 'ones' place.
• Second from the right is the 'tens' place.
• Third from the right is the 'hundreds' place.

In our addition problem, aligning digits into their respective place value columns makes adding extensive numbers manageable and accurate. Knowing place value also simplifies rounding numbers, as we'll see in the next section.

Rounding Rules

Rounding numbers helps simplify figures for easier comprehension or estimation, often necessary after complex operations like addition. When rounding to the nearest hundred, the key is to focus on the tens digit of the number. Following these rounding rules:

• If the tens digit is 5 or greater, round up by adding 1 to the hundreds place.
• If it is less than 5, round down, meaning the hundreds digit stays the same and all following digits turn to zero.

For our summed number, 233,433, check the tens digit, which is 3. Since 3 is less than 5, we round down, resulting in the rounded figure 233,400. This manipulation helps create more work-friendly numbers while retaining a degree of accuracy.