Problem 62 - Prealgebra

Question

Add or subtract the decimals, as indicated. \(65.079-(-52.6)\)

Step-by-Step Solution

Answer

117.679

Step 1: Understand the Expression

The expression given is \(65.079 - (-52.6)\). Notice that subtracting a negative number is equivalent to adding the positive of that number.

Step 2: Rewrite the Expression

Since subtracting a negative is the same as adding, rewrite the expression as: \(65.079 + 52.6\).

Step 3: Align the Decimals

Line up the decimal points of the numbers for easy addition: \(\begin{array}{r} 65.079 \+52.600 \\hline\end{array}\)

Step 4: Add the Numbers

Add the numbers starting from the rightmost digit, moving to the left, carrying over when necessary. \(\begin{array}{r} 65.079 \+52.600 \\hline 117.679 \\end{array}\)

Step 5: Write the Final Answer

The result of \(65.079 - (-52.6)\) is \(117.679\).

Key Concepts

Adding Decimals

When adding decimals, the key element is to ensure that the numbers are correctly aligned at the decimal point. This alignment is fundamental as it guarantees that you're adding the right digits together, according to their place value. For example, in the expression \(65.079 + 52.6\), first rewrite 52.6 as 52.600 to have the same number of decimal places. This makes it easier to align and add. Remember:

• Each digit's place in terms of tenths, hundredths, etc., has to be respected.
• Fill any missing decimal places with zeros if needed.

After aligning the decimals, proceed to add the numbers just like you would in normal addition, starting from the rightmost digit and moving leftwards. Whenever a column sums to more than 9, carry over the extra value to the next column to the left, just as with whole numbers. By aligning the decimal points and adding carefully, you ensure accuracy in your sum.

Subtracting Negative Numbers

Subtracting a negative number might seem tricky at first, but it's actually a straightforward concept once you understand it. In essence, subtracting a negative is the same as adding a positive. This occurs because two negatives make a positive.

• For example, the expression \(65.079 - (-52.6)\) transforms into \(65.079 + 52.6\).
• Think of it as canceling out the negative sign and replacing the operation with a plus.

Visualizing this process as moving on a number line can be helpful: you're effectively adding the distance, moving to the right (positively) by the value of the negative number. Once you make this conversion, proceed with the addition as usual. Understanding this concept allows you to tackle any math problem involving the subtraction of negatives with confidence.

Aligning Decimal Points

Aligning decimal points is a critical step in performing operations with decimal numbers. It's essential for maintaining consistency in the place values, ensuring each digit is added or subtracted in the correct column.

• When adding or subtracting decimals, line up the decimal points vertically.
• Use zeros to fill in any missing decimal places so that all numbers have the same length of decimal parts.

For example, in the calculation of \(65.079 + 52.6\), aligning means rewriting the numbers as:\[\begin{array}{r}65.079 \+52.600 \\hline\end{array}\]This method avoids errors that might stem from unmatched decimal places. By aligning the decimal points, you preserve the integrity of each digit's value, allowing for precise arithmetic operations. This is applicable for both addition and subtraction of decimals.