Problem 65 - Prealgebra

Question

Multiply. $$-\frac{1}{3}(-3)$$

Step-by-Step Solution

Answer

The product of \(-\frac{1}{3}(-3)\) is 1.

Step 1: Identify the Numbers

We are tasked to multiply two numbers: \(-\frac{1}{3}\) and \(-3\). The first number is a negative fraction, and the second is a negative integer.

Step 2: Multiply the Absolute Values

To multiply the two numbers, first find the product of their absolute values: \(\frac{1}{3} \times 3 = 1\).

Step 3: Determine the Sign of the Product

Since both numbers are negative, the product of two negative numbers is positive. Therefore, the result of \(-\frac{1}{3}(-3)\) gives a positive product.

Step 4: Combine Results

Combine the results from the previous steps: the absolute value of the product is 1, and the sign is positive. Thus, \(-\frac{1}{3}(-3) = 1\).

Key Concepts

Multiplying Negative Numbers

When multiplying negative numbers, it might seem a bit confusing initially, but understanding a few simple rules can make it much clearer. Imagine you have two numbers \(-\frac{1}{3}\) and \(-3\). Both of these numbers are negative. The question is what happens when you multiply them together?Here's what happens:

• If you have two negative numbers, and you multiply them, the result will always be positive.
• The reason for this can be thought of as "two wrongs make a right." In mathematics, if you "flip" twice, you end up back where you started, which is a positive place!

This is part of what leads us to see that \(-\frac{1}{3}(-3) = 1\). Understanding the multiplication of negatives is essential in algebra and ensures you finish with correct answers.

Absolute Value Product

In mathematics, the absolute value of a number is its distance from zero on the number line, without considering the direction. When multiplying, the absolute value product concept becomes handy. To find the absolute value of a number:

• Ignore the sign of the number. For example, the absolute value of \(-3\) is \3\, and the absolute value of \(-\frac{1}{3}\) is \frac{1}{3}\.
• Multiply these absolute values. So, for \(-\frac{1}{3}\) and \(-3\), multiply as \frac{1}{3} \times 3 = 1\.

This step helps ensure you correctly calculate the size of your answer, without worrying about the positive or negative sign at first. It's about focusing on the numbers themselves.

Sign Rules for Multiplication

The sign rules for multiplication not only apply to numbers in prealgebra but are foundational for everything that follows. Here are the key rules:

• **Rule 1:** Positive \times Positive = Positive. If both numbers are positive, the product is positive.
• **Rule 2:** Negative \times Negative = Positive. Just like with \(-\frac{1}{3}(-3)\), two negatives multiply to make a positive.
• **Rule 3:** Positive \times Negative = Negative. One positive and one negative number will result in a negative product.

When you apply these rules, it becomes easy to determine the sign of the result without second-guessing. Simply remember, count the negative numbers: if you have an even number of negatives, the product is positive; if odd, it's negative.