Problem 55 - Prealgebra

Question

Add or subtract the decimals, as indicated. \(-6.32+(-48.663)\)

Step-by-Step Solution

Answer

The result is -54.983.

Step 1: Recognize the Operation

The expression given is \[-6.32 + (-48.663)\]which involves the addition of two negative decimal numbers.

Step 2: Understand Negative Addition

When adding two negative numbers, the result is negative. Therefore, the operation essentially involves adding the absolute values of these numbers and attaching a negative sign to the result.

Step 3: Align Decimal Points

Rewrite the numbers vertically, aligning the decimal points:\[\begin{array}{r}-6.320 \-48.663\end{array}\]

Step 4: Add the Absolute Values

Add the absolute values of these numbers:\[\begin{array}{r}\phantom{-} 6.320 \+48.663\hline54.983\end{array}\]

Step 5: Apply the Negative Sign

Since both original numbers were negative, the sum is also negative:\[-54.983\]

Key Concepts

Subtracting Decimals

When subtracting decimals, the key idea is to find the difference between two numbers that have decimal values. Often, this operation requires attention to detail, especially when dealing with negative numbers or numbers that have more decimal places. Subtracting one negative number from another might involve additional steps, such as forming equivalent positive expressions with absolute values. For example, when you have to subtract negative numbers with decimals, you first need to consider the rules of integers.

• Subtracting a negative is the same as adding its positive counterpart.
• For example, subtracting \(-6.32\) from \(-48.663\) is similar to adding their absolute values due to negatives canceling out negatives.

By knowing these basic principles, you can easily manage more complex expressions and correctly determine the sign and magnitude of the result.

Aligning Decimal Points

Aligning decimal points is crucial when adding or subtracting decimal numbers. This process ensures that each digit is combined with its proper positional neighbor, based on its place value, which maintains mathematical accuracy and clarity.To align decimal points properly:

• Write the numbers one under the other, aligning the decimal points vertically.
• Fill in any empty decimal places with zeros to ensure each number has the same number of decimal places. For example, transform \(-6.32\) to \(-6.320\) when aligning with \(-48.663\).

By aligning the decimals, you can more easily perform the arithmetic operations, ensuring all numbers are calculated correctly and efficiently. More so, this step reduces future mistakes when simplifying or solving complex calculations. Aligning decimal points becomes especially important when numbers are heavily decimal-based and need precise handling.

Absolute Values of Decimals

Understanding the absolute values of decimals is pivotal when dealing with both addition and subtraction involving negative numbers. The absolute value is essentially the magnitude of a number regardless of its sign, giving you a clear picture of its size compared to other numbers.With decimals, this concept helps simplify operations involving negative numbers:

• Take the number without considering its sign, as this represents its absolute value.
• For example, the absolute value of \(-6.32\) is \(6.32\).

Knowing the absolute values allows more straightforward calculation steps. If both numbers are negative, as in the example of \(-6.32+(-48.663)\), find each one's absolute value. Then add these values together to get the sum before applying the negative sign back, ensuring you retain the correct overall negativity of the original numbers. This approach helps in accurately calculating sums or differences, reducing errors during such processes.