Problem 72 - Prealgebra

Question

Simplify the given expression. \(-8.14+(16.6-(-15.41))\)

Step-by-Step Solution

Answer

The simplified expression is 23.87.

Step 1: Identify the Operation inside the Parentheses

Begin by looking at the expression inside the parentheses: \((16.6 - (-15.41))\). Notice that when you subtract a negative number, it is equivalent to adding the absolute value of that number.

Step 2: Simplify the Expression inside the Parentheses

Convert the subtraction of a negative number into addition: \(16.6 - (-15.41) = 16.6 + 15.41\).

Step 3: Perform the Addition

Add the numbers inside the parentheses: \(16.6 + 15.41 = 32.01\). This simplifies the original expression to \(-8.14 + 32.01\).

Step 4: Perform the Final Addition

Add \(-8.14\) to \(32.01\): \(-8.14 + 32.01 = 23.87\). This is the simplified result of the original expression.

Key Concepts

Understanding Operations with Negative Numbers

Operations with negative numbers might seem tricky at first, but with a little practice, they become much more manageable. Negative numbers are numbers that are less than zero. When dealing with negative numbers, one important rule to remember is that subtracting a negative number is the same as adding its positive counterpart.

• For instance, when we have \(-(-5)\), it is equivalent to \(+5\).
• This is because subtracting something negative "undoes" the negativity, effectively turning it positive.

Always double-check the signs when performing operations involving negative numbers. It's crucial to maintain accuracy since a small sign error can lead to different results.

Using Parentheses in Arithmetic

Parentheses in arithmetic dictate the order in which operations are performed. They help group parts of an expression to ensure that calculations are done correctly and in the correct order. Think of parentheses like a signal to do those operations first.

• In expressions, always tackle operations inside parentheses before anything else.
• Understand that nested parentheses (parentheses within parentheses) should be solved from the innermost set to the outer.
• Remember the basic rule: Perform operations inside parentheses before moving on to addition, subtraction, etc.

In our example, the expression \((16.6 - (-15.41))\) required simplification first, and only then could we proceed to the next operations outside the parentheses.

Mastering Addition with Decimals

Addition with decimals is essential and can be mastered with practice. When adding decimals, ensure that their decimal points line up vertically. This ensures you are adding digits of equal place value, such as tenths with tenths, hundredths with hundredths, etc.

• Align the numbers by their decimal points, not by their edges.
• Fill in with zeros if needed, so each number has the same number of decimal places. For example, treat \(5.6\) as \(5.60\) when adding it to \(6.78\).
• Proceed to add as you would whole numbers, starting from the rightmost digit.

In our case, correctly aligning and adding \(16.6\) and \(15.41\) results in \(32.01\). This method ensures accurate calculations and problem-solving with decimals.