Problem 19 - Elementary Algebra

Question

Find the product of 5 and -12 .

Step-by-Step Solution

Answer

The product of 5 and -12 is -60.

Step 1: Identify the Numbers

In this exercise, we need to multiply two numbers: 5 and -12.

Step 2: Understand Multiplication with Negative Numbers

When multiplying two numbers where one number is negative, the product will be negative. This is because multiplying by a negative number reverses the sign.

Step 3: Multiply the Absolute Values

First, ignore the signs and multiply the absolute values of the numbers: 5 and 12. So, calculate: \[5 \times 12 = 60\]

Step 4: Apply the Negative Sign

Since one of the numbers is negative, the product will be negative. Thus, the result is: \[-60\]

Key Concepts

Understanding Negative Numbers

Negative numbers can be a bit tricky, especially when you first encounter them in multiplication. These are numbers less than zero, and they are typically written with a minus sign (-) in front. When you multiply by a negative number, you're essentially reversing the direction on a number line. That means your positive values become negative and vice versa.

This reversal happens because multiplying by a negative number flips the sign of the product. A handy rule to remember is:

• Positive imes Positive = Positive
• Positive imes Negative = Negative
• Negative imes Positive = Negative
• Negative imes Negative = Positive

Being comfortable with these rules will help you easily work through problems involving negative numbers.

Grasping Absolute Values

The absolute value of a number refers to its distance from zero on the number line, without considering which direction from zero the number lies. Essentially, absolute value is all about magnitude, ignoring the sign of the number. For instance, the absolute value of both -12 and 12 is 12 because they are 12 units away from zero.

When it comes to multiplication, especially with negatives, it's useful to first consider the absolute values of the numbers. This approach simplifies calculation by temporarily ignoring the negative signs. Once you've multiplied the absolute values, then use the rules about negative signs to determine the final sign of your product.

• Absolute value of 5: 5
• Absolute value of -12: 12

Multiply them, as in the given problem, and then apply the appropriate sign to your result.

Breaking Down Step-by-Step Solutions

Breaking down problems into a step-by-step format can make tough math problems easier to handle. In the given example, the task was to find the product of 5 and -12. Let's see how we can manage through a detailed approach:

• Identify the numbers: Start with recognizing the numbers involved, which are 5 and -12 in this case.
• Understand the signs: Recognize that one number is negative and understand the implications on the result.
• Multiply the absolute values: Ignore the sign temporarily and multiply the absolute values:
• Apply the negative sign: Finally, apply the negative sign where needed, concluding with the result of -60.

By following these structured steps, students can repeat the process for different numbers and enhance their understanding of multiplication involving both positive and negative numbers.