Problem 33 - Prealgebra

Question

Add. $$3,210+224,7$$

Step-by-Step Solution

Answer

The sum is 3,434.7.

Step 1: Align the Numbers

First, we need to align the numbers according to their place values so that the decimal points are aligned. Write \( 3,210 \) as \( 3,210.0 \), and then place \( 224.7 \) under it.

Step 2: Perform the Addition

Start adding from the rightmost column (the tenths place) and move to the left. Add each pair of digits, carrying over if necessary:\[\begin{align*} &3,210.0 \ +&\phantom{0}224.7 \ \hline &3,434.7 \ \end{align*}\]

Step 3: Combine the Results

Notice the addition results in \( 3,434.7 \). Since there are no more digits to add, this is the final sum.

Key Concepts

Place Value

When you're performing addition, understanding place value is crucial. Place value refers to the value of a digit depending on its position in the number. For instance, in the number 3,210, each digit represents a different power of ten:

• 3 is in the thousand's place, meaning it represents 3,000.
• 2 is in the hundred's place, standing for 200.
• 1 is in the ten's place, which makes it 10.
• 0 is in the one's place, amounting to zero.

The same concept applies to the number 224.7:

• 2 represents 200 (hundred's place).
• 2 is in the ten's place, meaning 20.
• 4 is in the one's place, meaning 4.
• 7 is in the tenth's place, representing 0.7.

When adding numbers, it's important that digits occupy the correct place values to get an accurate result.

Alignment

Alignment in addition is like lining up the players on a sports team. Each player knows their position, ensuring the team functions effectively. Similarly, aligning numbers correctly, particularly with decimals, ensures every digit is added to its rightful counterpart.
To align numbers such as 3,210 and 224.7 for addition, you treat 3,210 as 3,210.0. This maintains the integrity of each place value by ensuring all columns line up neatly.
This organization is key because it:

• Prevents calculation mistakes by keeping digits in their respective slots.
• Keeps decimal points aligned, especially crucial when dealing with mixed numbers (whole numbers and decimals).
• Ensures carries and borrows happen in the correct position if necessary.

Practicing alignment will boost your confidence anytime you need to sum numbers.

Decimal Addition

Decimal addition might seem intimidating at first, but it's simple if you keep a few pointers in mind. Start from the rightmost digit, exactly as you would when adding whole numbers. In the case of 3,210.0 and 224.7:

• Start adding from the tenths position, move to the one's, ten's, hundred's, and so on.
• If the sum of a column exceeds 9, carry over to the next column on the left. For instance, if a column adds up to 14, write down 4 and carry 1 to the next column.

An important aspect of decimal addition is keeping the decimal points in line. This maintains the integrity of the place values. After completing the sum, double-check that the decimal point in the answer reflects the right position just as in the original numbers.
By practicing decimal addition regularly, it becomes as intuitive as whole number addition. Remembering these tips will make tackling decimal numbers a lot more manageable.