Problem 66 - Prealgebra

Question

Multiply. $$-\frac{1}{4}(-4)$$

Step-by-Step Solution

Answer

1

Step 1: Identify the Signs

Identify that you are multiplying a negative number \(-\frac{1}{4}\) by another negative number \(-4\). Recall that the product of two negative numbers is positive.

Step 2: Multiply Numerical Values

Multiply the absolute values of the numbers involved: \( \frac{1}{4} \times 4 \). This is equivalent to multiplying \(1\) by \(4\), then dividing the result by \(4\).

Step 3: Simplify the Multiplication Result

Calculate \( \frac{1}{4} \times 4 = 1 \) because \(4 \div 4 = 1\).

Step 4: Determine the Final Sign

Since the multiplication involved two negative numbers, the result is positive. Thus, the final result is \(1\).

Key Concepts

Multiplying Negative Numbers

When you multiply two negative numbers, your result is always positive. But why is that? Think of multiplying as repeated addition. If we consider subtracting a negative as adding a positive, then multiplying two negatives becomes similar. Here's a simple way to understand it:

• Multiplying a negative by a positive number: You effectively add the negative repeatedly, so the answer stays negative.
• Multiplying two negative numbers: Each negative number 'cancels' out, turning the situation around, resulting in a positive outcome. This is like going backwards twice.

Now remember, once you recognize that both multipliers are negative, you can immediately know the result is positive, no matter what numbers you're working with. This makes the task of solving equations quicker and easier.

Absolute Value

The concept of absolute value helps simplify operations with negative numbers. The absolute value of a number is its distance from zero on a number line, regardless of its direction. It is always a non-negative value. For instance, the absolute value of \(-4\) is \(4\), and the absolute value of \(\frac{1}{4}\) is still \(\frac{1}{4}\).When solving multiplication problems involving negative numbers, focusing on their absolute values can make the arithmetic simpler:

• Strip away the negative signs and multiply as if they are both positive numbers.
• Calculate the final product using these absolute values.
• Use the rules about negative multipliers to determine the sign of your result.

This approach streamlines calculating complex expressions and helps avoid sign errors.

Multiplication Rules in Prealgebra

In prealgebra, understanding multiplication rules is crucial. These rules help you work not just with positive numbers, but also with negatives and fractions:

• Positive × Positive = Positive: Simple enough, two positives make it bigger.
• Negative × Negative = Positive: Remember, two negatives cancel each other out.
• Positive × Negative = Negative: If a positive and a negative number multiply, the result is negative.

For fractions, multiply straight across—numerator with numerator, denominator with denominator. For instance, \(\frac{1}{4} \times 4\) means you just multiply the top number 1 by 4 and keep the denominator 4, resulting in \(\frac{4}{4} = 1\). Understanding these rules helps make finding the product much simpler and less intimidating.