Problem 78 - Prealgebra

Question

Simplify the given expression. \((-4.98)(-1.7)-3.52\)

Step-by-Step Solution

Answer

The simplified expression is 4.946.

Step 1: Multiply the Negatives

First, multiply the two negative numbers inside the parentheses: \((-4.98) \times (-1.7)\).Since the multiplication of two negative numbers results in a positive number, perform the multiplication as follows:\[4.98 \times 1.7 = 8.466.\]Therefore, the result is positive 8.466.

Step 2: Subtract the Constant

After obtaining the result from step 1, we now need to subtract 3.52 from 8.466:\[8.466 - 3.52 = 4.946.\]This completes the simplification of the expression.

Key Concepts

Negative Numbers

Negative numbers behave differently in mathematics compared to positive numbers. Instead of merely having a value, they have a value less than zero. They are represented with a minus sign, for example, -1, -5, or -3.2. In real-world terms, you can think of negative numbers as a debt or loss. An essential rule when working with negatives is understanding multiplication involving negative numbers.

• When two negative numbers are multiplied, the result is positive. This might seem counterintuitive at first, but it's due to the negative symbol denoting an opposite direction or "flipping around" the number line.
• If a negative number is multiplied by a positive number, the result is negative, resembling a change in direction towards the opposite side on a number line.

In the exercise given, \((−4.98) \times (−1.7)\), both numbers are negative. So, the product turns positive, resulting in 8.466.

Multiplication

In mathematics, multiplication is simply the process of adding a number to itself a certain number of times. When multiplying whole numbers, it’s straightforward. However, with decimals and negatives, extra care is needed.

• Ensure alignment of decimal points. Although they are not visible, they impact calculation steps significantly in multiplication.
• Ignore the signs and initially treat the numbers as positives to simplify the calculation, then apply the rule for signs at the end.
• If both numbers have more than one decimal, it’s crucial to multiply them as whole numbers first, count the decimal places in total from both numbers, and place the decimal accordingly after multiplying.

For instance, in \((-4.98) \times (-1.7)\), multiplying without concern for the negative signs initially gives you 8.466. The signs are then assessed, resulting in a positive outcome as both factors were negative.

Subtraction

Subtraction is another fundamental operation, often considered the inverse of addition. When subtracting numbers, you’re essentially finding the difference between them. Here are some key points to make subtraction clearer:

• Always arrange numbers correctly to avoid mistakes, especially aligning decimal points with decimal numbers.
• Check your borrowing technique. Often with decimals, neglecting to "borrow" from the next higher place value can lead to errors, just like in whole numbers.
• When dealing with negative numbers, a common scenario is to transform subtraction into the addition of a negative, which sometimes simplifies understanding. For instance, subtracting a negative equates to adding a positive.

In the example \(8.466 - 3.52\), ensure the decimal places align to get the difference 4.946 efficiently. Each concept builds on this essential skill, impacting correctness in simplification tasks.