Problem 20 - Beginning and Intermediate Algebra

Question

Multiply. $$ -0.5(-0.3) $$

Step-by-Step Solution

Answer

The product is 0.15.

Step 1: Understand Multiplication of Negatives

The first step is to understand the rule of multiplying negative numbers. When you multiply two negative numbers, the result is positive. This is because a negative multiplied by a negative makes a positive.

Step 2: Multiply the Absolute Values

Ignore the negative signs for a moment and simply multiply the absolute values of the numbers. The absolute value of \(-0.5\) is \(0.5\), and the absolute value of \(-0.3\) is \(0.3\). So, \(0.5 \times 0.3 = 0.15\).

Step 3: Apply the Sign Rule

Since both numbers in the multiplication were negative, we now apply the rule for negative multiplication: the result is positive. Therefore, the answer to \(-0.5 \times -0.3\) is \(0.15\).

Key Concepts

Absolute Values

Absolute values are crucial in mathematics because they simplify and streamline many operations. The absolute value of a number is its distance from zero on the number line, without considering direction. In simpler terms, it's always a non-negative number, typically expressed as \(|x|\) where \(x\) is the given value. For example:

• The absolute value of \(-0.5\) is \(0.5\).
• The absolute value of \(-0.3\) is \(0.3\).

When solving problems that involve multiplication of negative numbers, we often first consider their absolute values. This approach helps to disregard the potential confusion caused by negative signs, focusing on pure numerical computation instead. Once you have computed the basic product of these absolute values, you can reinstate the appropriate sign to find the final solution.

Positive and Negative Rules

Understanding the rules of multiplying positive and negative numbers is fundamental in arithmetic. Here are some simple guidelines to remember:

• Multiplying two positive numbers results in a positive number \( (3 \times 2 = 6) \).
• Multiplying two negative numbers results in a positive number \((-0.5 \times -0.3 = 0.15)\).
• Multiplying a positive and a negative number results in a negative number \( (4 \times -1 = -4) \).

These rules stem from how multiplication acts as repeated addition in mathematics. If you think about a negative times a negative, you are essentially reversing the direction of each negative factor, leading to a positive outcome. It's just like walking backward to move forward from a different perspective.

Step-by-Step Problem Solving

To solve multiplication problems, especially those with negative numbers, a step-by-step approach is very helpful. This helps to avoid mistakes and ensures a thorough understanding of the process. Let's break it down further:1. **Identify the Signs**: Examine the numbers you need to multiply. Determine if they are positive or negative.2. **Calculate Absolute Values**: Treat the negative numbers as if they are positive by using their absolute values. This is done to simplify the multiplication without worrying about the signs.3. **Conduct the Multiplication**: Multiply the absolute values together. For example, the absolute values \(0.5\) and \(0.3\) multiply to \(0.15\).4. **Apply the Sign Rules**: Reintroduce the correct sign to your final answer. Since multiplying two negatives gives a positive, the result of the example problem is positive \(0.15\).Approaching each problem with this systematic method can alleviate confusion and build confidence in your mathematical ability.