Problem 7 - Prealgebra

Question

Add the decimals. \(95.57+7.88\)

Step-by-Step Solution

Answer

The sum of 95.57 and 7.88 is 103.45.

Step 1: Align the Decimals

Begin by aligning the numbers by their decimal points: \[\begin{array}{c}95.57 \+ 7.88 \\hline\end{array}\] This makes it easier to add each column correctly.

Step 2: Add the Hundredths Column

Add the digits in the hundredths place (farthest right): 7 + 8 = 15. Place 5 in the hundredths place of the sum and carry over 1 to the tenths column.

Step 3: Add the Tenths Column

Add the digits in the tenths place along with the 1 carried over: 5 + 8 + 1 = 14. Place 4 in the tenths place of the sum and carry over 1 to the units column.

Step 4: Add the Units Column

Add the units digits and the 1 carried over: 5 + 7 + 1 = 13. Place 3 in the units column and carry over 1 to the tens column.

Step 5: Add the Tens Column

Finally, add the digits in the tens place along with the carried over 1: 9 + 1 = 10. Place 0 in the tens place and 1 in a new hundreds place.

Step 6: Write the Final Sum

Compile the results from each column to form the final sum:\[\begin{array}{c}103.45\end{array}\]

Key Concepts

Decimal Alignment

When adding decimals, aligning the numbers correctly is crucial. This alignment ensures each digit is in the proper column, corresponding to its place value. Whether you're dealing with whole numbers or decimals, always line up the decimal points vertically.
This way, each column represents a specific place value, such as tenths, hundredths, or units. By maintaining this order, you reduce the chances of errors when adding the numbers. Misalignment can lead to incorrect sums because the numbers aren't matched per their place value. Proper alignment is the foundation for accurate decimal addition.

Carrying Over

Carrying over is an essential step in addition, especially when results exceed the base 10 in one column. In the context of adding decimals, carrying over involves moving the extra value to the next column on the left. For instance, if adding two numbers gives a sum greater than 9, you place the digit representing the units in the current column and carry over the tens digit to the next column.

• Hundredths: 7 + 8 = 15 → Place 5, carry 1 to tenths.
• Tenths: 5 + 8 + 1 = 14 → Place 4, carry 1 to units.
• Units: 5 + 7 + 1 = 13 → Place 3, carry 1 to tens.

Understanding carrying over is key to mastering addition, especially when working with multiple columns.

Place Value

Place value is a fundamental concept for adding decimals. Each number holds a specific position based on its distance from the decimal point. These positions are:

• Tens
• Units
• Tenths
• Hundredths

Recognizing the place value ensures that each digit is added correctly within its column. The position of each digit determines its value, such as how 9 in the tens place represents 90 while 9 in the units place represents simply 9. Place value provides a structured approach to addition, simplifying complex problems by breaking them into manageable parts.

Step by Step Addition

Using a step-by-step approach in decimal addition allows for clarity and precision. This methodical process helps minimize mistakes during calculation. Let's review a typical sequence when adding decimals: 1. **Align the decimals**: Ensure that all decimal points form a straight, vertical line. 2. **Add smallest place value**: Begin from the right with the smallest place value, the hundredths. 3. **Carry over when needed**: If sums exceed 9, carry the excess to the next higher column. 4. **Continue leftwards**: Move from hundredths to tenths, units, tens, and so forth.
This method guarantees that each step is performed with attention to detail, leading to an accurate final result. By breaking down quickly overwhelming calculations into smaller, easier parts, students gain confidence and accuracy in their mathematical operations.