Problem 28 - Prealgebra

Question

Multiply each of the following. Be sure all answers are written in lowest terms. $$-\frac{1}{5}(-5)$$

Step-by-Step Solution

Answer

The product of \(-\frac{1}{5}(-5)\) is 1.

Step 1: Apply the Multiplication Rule for Fractions

To multiply a fraction by a whole number, multiply the numerator of the fraction by the whole number, while keeping the denominator the same.

Step 2: Identify the Numbers Involved

We are given the expression \(-\frac{1}{5} \times (-5)\). Here, the fraction is \(-\frac{1}{5}\) and the whole number is \(-5\).

Step 3: Multiply the Numerator

Multiply the numerator \(-1\) of the fraction by the whole number \(-5\): \(-1 \times (-5) = 5\).

Step 4: Retain the Denominator

Keep the denominator as it is: 5. Therefore, the product becomes \(\frac{5}{5}\).

Step 5: Simplify the Fraction

Simplify \(\frac{5}{5}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 5. So, \(\frac{5}{5} = 1\).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential step to ensure that the fraction is expressed in its simplest form. Consider the fraction \( \frac{a}{b} \), where \( a \) is the numerator and \( b \) is the denominator. A fraction is simplified when the only common factor between \( a \) and \( b \) is 1.

• To simplify, divide both the numerator and the denominator by their greatest common divisor (GCD).
• If the GCD of \( a \) and \( b \) is 1, the fraction is already in simplest form.
• In our example, \( \frac{5}{5} \), both the numerator and the denominator can be divided by 5, which simplifies to 1.

Simplified fractions are often more intuitive and easier to manipulate in further calculations.

Multiplication Rules for Fractions

When multiplying fractions and whole numbers, understanding the multiplication rules can simplify the process. Here's how to handle such multiplications:

• Multiply the numerators together. If one of the numbers is a whole number, treat it like a fraction with denominator 1.
• Multiply the denominators together. If you have a whole number, its denominator is essentially 1.
• In our exercise, we worked with \( -\frac{1}{5} \times (-5) \). We multiply the numerator \(-1\) by the whole number \(-5\), resulting in 5, and keep the denominator as 5.

The product of the fractions results in a new fraction, which may require simplification to its lowest terms.

Absolute Value in Multiplication

Absolute value comes into play when dealing with negative numbers, particularly in multiplication. The absolute value of a number is the distance from zero on the number line, always expressed as a non-negative value.

• To find the absolute value, strip away the negative sign. For example, the absolute value of \(-5\) is 5.
• When multiplying negative numbers, remember that the product of two negative numbers is a positive number.
• Thus, both \(-1\) and \(-5\) in our example will contribute to making the result positive 5, as their absolute values make the rule \((-) \times (-) = +\) true.

Understanding absolute values helps eliminate confusion about sign changes, ensuring accurate arithmetic operations.