Problem 43 - Prealgebra

Question

Add or subtract the decimals, as indicated. \(71 .72-(-6)\)

Step-by-Step Solution

Answer

The result of the operation \(71.72 - (-6)\) is 77.72.

Step 1: Understand the Operation

The expression given is \(71.72 - (-6)\). Here, you are subtracting a negative number from a decimal. Remember that subtracting a negative is equivalent to adding a positive.

Step 2: Convert the Subtraction to Addition

Convert the expression \(71.72 - (-6)\) to addition. This changes the expression to \(71.72 + 6\).

Step 3: Align the Decimals

Before performing the addition, write 71.72 and 6 below each other, aligning their decimal points:\[\begin{align*} 71.72 \+ 6.00 \end{align*}\]

Step 4: Perform the Addition

Add the numbers as you normally would: - Add the hundredths place: 2 + 0 = 2. - Add the tenths place: 7 + 0 = 7. - Add the units place: 1 + 6 = 7. - Add the tens place: 7 + 0 = 7. This gives you a result of 77.72.

Step 5: Confirm the Result

Double-check your addition to ensure there are no errors. After verification, the final result obtained is correct at 77.72.

Key Concepts

Subtracting Negative Numbers

Subtracting a negative number can be a tricky concept initially. However, it is essential to understand because it frequently appears in arithmetic.
Imagine you have an equation like this: \(a - (-b)\). In such cases, subtracting the negative number \(-b\) translates into adding a positive number \(+b\). This is because two negative signs cancel each other out, turning the operation into addition.
For example, in the problem: \(71.72 - (-6)\), we change it to \(71.72 + 6\). Once you grasp this concept, subtracting negative numbers becomes just as straightforward as a typical addition.

Alignment of Decimals

Proper alignment of decimal points is crucial when adding or subtracting decimal numbers. It ensures that numbers in each decimal place are aligned with their corresponding places in other numbers (units with units, tenths with tenths, etc.).
Essentially, line up the numbers so their decimal points form a vertical line. This way, each digit is in the right column, allowing for accurate addition or subtraction.
For instance, consider aligning 71.72 and 6. Even though 6 is a whole number, we can consider it as 6.00 to make the decimals align perfectly:

• Write 71.72
• Place 6.00 beneath it

Now, adding becomes more manageable as the digits are neatly aligned.

Arithmetic Operations with Decimals

Performing arithmetic operations on decimals involves a few more steps than with whole numbers but follows similar basic principles.
Once the decimal points are aligned accurately, you proceed with addition starting from the rightmost digit and moving towards the left. Add digits in same places just like in whole numbers. Be sure to carry over any overflow if a column sums beyond 9.
In our example, adding:

• Hundredths: 2 + 0 = 2
• Tenths: 7 + 0 = 7
• Units: 1 + 6 = 7
• Tens: 7 + 0 = 7

Thus, the sum of \(71.72 + 6\) results in 77.72. You can then verify for any simple errors but you can be confident when your decimals are properly aligned.