Problem 13 - Prealgebra

Question

Find each of the following sums. (Add.) $$\begin{array}{l}7.123 \\\8.12 \\\9.1\end{array}$$

Step-by-Step Solution

Answer

The sum is 24.343.

Step 1: Align the Numbers

Firstly, write each number one below the other, aligning the decimal points. This helps in ensuring that each digit is in the correct place value:\[\begin{array}{r}7.123 \8.120 \9.100 \\hline\end{array}\]

Step 2: Add the Thousandths Place

Starting from the rightmost column, add the digits in the thousandths place. Since 8.120 and 9.100 have no digits in the thousandths place, assume they have zeroes. The calculation is:\[3 + 0 + 0 = 3\]

Step 3: Add the Hundredths Place

Move to the hundredths place and add the numbers in this column:\[2 + 2 + 0 = 4\]

Step 4: Add the Tenths Place

Add the digits in the tenths place:\[1 + 1 + 1 = 3\]

Step 5: Add the Units Place

Add the digits in the units place:\[7 + 8 + 9 = 24\]. Write down 4 in the unit's place and carry over 2 to the left side.

Step 6: Account for the Carry-over

Since there was a carry-over from the units place, add 2 to the next column (the tens place). If there are no digits remaining, simply write down the carry-over as the last part of the sum.

Step 7: Write Down the Final Sum

Combine all the results from the above steps to write down the final sum:\[\begin{array}{r}7.123 \8.120 \9.100 \\hline24.343 \\end{array}\]

Key Concepts

Understanding Place Value

Understanding the concept of place value is essential for dealing with decimal numbers. Each digit in a decimal number has a specific position that determines its value. The place value depends on how far the digit is from the decimal point. Moving from left to right:

• The first digit to the left of the decimal point is the units place.
• The next digit to the left is the tens place, and further left is the hundreds place, and so on.
• The first digit to the right of the decimal point is the tenths place.
• The next digit after that is the hundredths place, continuing with the thousandths place.

In the issue of adding 7.123, 8.12, and 9.1, being aware of place value assists in ensuring each digit aligns with a number of similar value, securing correctness in the addition process.
For instance, the digit 7 in 7.123 represents seven units (or ones), while the 1 in 7.123 represents one-tenth. This clarity conserves precision and accuracy when performing operations like addition.

Aligning Decimals for Addition

When adding decimal numbers, aligning the decimal points is crucial as it helps maintain place value, which is fundamental in performing accurate addition. Here's a quick guide on how to align the decimals:

• Start by writing the decimal numbers one on top of the other.
• Ensure that the decimal points are in a vertical line.
• This also aligns the ones, tens, and other place values accordingly, making it easier to add vertically.
• Fill any empty spaces with zeroes to balance the place values across different numbers.

For example, in the numbers 7.123, 8.12, and 9.1, aligning their decimal points ensures that each digit is added to its corresponding place value. To illustrate, the numbers could be adjusted to look like:

• 7.123
• 8.120
• 9.100

Once aligned correctly, you can begin adding from the rightmost column without fear of mismatched values.

Mastering Carrying Over in Addition

Carrying over is a vital technique used in addition when the sum of digits in any column surpasses 9. This is also applicable when adding decimals. Here's how to handle carry-over effectively:

• Begin the addition from the rightmost side (decimals), moving towards the left.
• If a column sum is 10 or more, write down the rightmost digit and carry the left digit to the next column.
• Continue this process, ensuring to account for any carry-over as you move to the left.

In the given exercise, after adding the units place (7 + 8 + 9), you get 24. You write down 4 and carry over 2 to the next left column. There aren't any digits left in this problem, so simply write down the carry-over 2 as part of the final sum, resulting in 24.343. Learning to manage carry-overs is pivotal in ensuring the precision of your addition.