Problem 27 - Elementary and Intermediate Algebra

Question

Multiply. $$ -5 \cdot(-2.3) $$

Step-by-Step Solution

Answer

11.5

Step 1: Understand the Problem

Identify the numbers you need to multiply. In this problem, you need to multiply $$-5$$ and $$-2.3$$.

Step 2: Use the Multiplication Rules for Negative Numbers

Remember that the product of two negative numbers is positive. So, the sign of the result will be positive.

Step 3: Perform the Multiplication

Ignore the signs and multiply the absolute values of the numbers: $$5 \times 2.3 = 11.5$$.

Step 4: Apply the Sign

Since the product of two negative numbers is positive, the final answer is $$11.5$$.

Key Concepts

Negative Multiplication Rules

Multiplying negative numbers can sometimes be tricky. However, remembering a few key rules can help a lot. When you multiply two negative numbers, their product is always positive. This is because negative signs cancel each other out. For instance, multiplying \( -5 \) and \( -2.3 \) follows this rule. Here's the reasoning:

• A negative times a negative equals a positive.
• A negative times a positive equals a negative.
• A positive times a positive equals a positive.

Simply put, when the number of negatives is even, the product is positive. When it is odd, the result is negative.

Absolute Value Multiplication

When multiplying negative numbers, it helps to focus on their absolute values first. The absolute value of a number is its distance from zero. For example, the absolute value of \( -5 \) is 5, and for \( -2.3 \), it is 2.3.
Here’s how you can perform the multiplication:

• Ignore the signs for a moment.
• Multiply the absolute values of the numbers.

In our example, \( |−5| = 5 \) and \( |−2.3| = 2.3 \). So, we multiply these values together:
\[ 5 \times 2.3 = 11.5 \]
This gives us the positive value of the multiplication. Keeping the rules of negative multiplication in mind, we then apply the correct sign to our result.

Positive Product of Negatives

The multiplication of two negative numbers results in a positive product. This concept might seem a bit counterintuitive, but it’s a fundamental rule of mathematics. Take the example of multiplying \( -5 \) and \( -2.3 \).

• First, we calculate the absolute values: 5 and 2.3.
• Multiply them: \[ 5 \times 2.3 = 11.5 \]
• Because both original numbers were negative, their product is positive: \( 11.5 \).

Keeping these steps in mind makes it much simpler to handle such multiplication problems. Remember, whenever you multiply two negative numbers, the strange-looking result is actually quite normal and always positive.