Problem 14 - Prealgebra

Question

Find each of the following sums. (Add.) $$\begin{array}{l}5.432 \\\4.32 \\\3.2\\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The sum is 12.952.

Step 1: Align the Numbers

Start by aligning the numbers vertically by their decimal points. This helps to ensure that digits with the same place value are added together. \[\begin{array}{r} 5.432 \+ 4.320 \+ 3.200 \\hline\end{array}\]

Step 2: Add the Thousandths

Begin with the thousandths column (the third decimal place from the left). Add the numbers vertically.\[2 + 0 + 0 = 2\]Write 2 in the thousandth place of the result.

Step 3: Add the Hundredths

Move to the hundredths column. Add the numbers in this column.\[3 + 2 + 0 = 5\]Write 5 in the hundredth place of the result.

Step 4: Add the Tenths

Now, add the tenths column numbers.\[4 + 3 + 2 = 9\]Write 9 in the tenth place of the result.

Step 5: Add the Units

Finally, add the units (whole numbers) column.\[5 + 4 + 3 = 12\]Write 2 in the units place and carry over 1 to the next column (tens place).

Step 6: Handle the Carry and Write the Result

Place the carryover from the previous step into the tens column. Since there are no other digits to add here, just write down the carried number.So, the sum is \[12.952\]

Key Concepts

Understanding Place Value in Decimal Addition

Place value is a fundamental concept in decimal addition that refers to the value of a digit depending on its position within a number. When dealing with decimals, the places are divided into whole numbers and decimals, separated by a decimal point.
For example, in the number 5.432, each digit has a specific place value:

• 5 is in the ones place.
• 4 is in the tenths place.
• 3 is in the hundredths place.
• 2 is in the thousandths place.

Recognizing these place values is crucial because it ensures that each digit is added to the correct corresponding place in the other numbers. By acknowledging each digit's place value, you prevent common mistakes in sum calculation.

Importance of Aligned Decimal Points

Aligned decimal points help you organize the digits by their place value for accurate addition. Properly aligning decimal points ensures that each column of digits represents the same place values like tenths, hundredths, and thousandths.

Consider the exercise where you have to add 5.432, 4.320, and 3.200. When you align these numbers by their decimal points, it looks like this:

• 5.432
• 4.320
• 3.200

This approach makes it easier to add each column separately, from right to left, without mixing the different place values, leading to a correct and precise sum.

The Role of Carrying Over in Decimal Addition

Carrying over happens when the sum of digits in a column exceeds nine. This concept is applied in both whole and decimal numbers. In decimal addition, when the digits are added up to ten or more, the extra value is carried over to the next column, increasing the total there.

In our example, adding the units (5 + 4 + 3) gives 12. The 2 is kept in the units place, and the 1, being extra, is carried over to the next column, the tens place. If this step wasn’t performed, the resulting sum would be incorrect.

• Ensuring the carried number is added to the next column maintains accuracy.
• Carrying over is essential when the sum of a column exceeds nine.

Remembering to carry over helps keep your calculations on track and prevents errors in your final answer.