Problem 35 - Elementary Algebra

Question

For exercises \(23-74\), evaluate. $$ \left(-\frac{1}{4}\right)\left(-\frac{5}{6}\right) $$

Step-by-Step Solution

Answer

\( \frac{5}{24} \)

Step 1: Recognize the Multiplication Problem

The task is to multiply two fractions: \ \ \( -\frac{1}{4} \) and \( -\frac{5}{6} \)

Step 2: Multiply the Numerators

Multiply the numerators of the two fractions: \( -1 \) and \( -5 \). \ \ \( -1 \times -5 = 5 \)

Step 3: Multiply the Denominators

Multiply the denominators of the two fractions: \( 4 \) and \( 6 \). \ \ \( 4 \times 6 = 24 \)

Step 4: Simplify the Fraction

The resulting fraction is \( \frac{5}{24} \). Since 5 and 24 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Fractions Multiplication

When multiplying fractions, the process is straightforward. You just follow these steps:
First, you multiply the numerators (the top numbers). Then, you multiply the denominators (the bottom numbers). This will give you a new fraction. So, for the exercise given, which involves multiplying \( -\frac{1}{4} \) by \( -\frac{5}{6} \), you would first multiply \( -1 \) by \( -5 \), resulting in \( 5 \). Then, you multiply \( 4 \) by \( 6 \), which equals \( 24 \). This leaves you with the fraction \( \frac{5}{24} \).
Here’s a quick summary:

• Multiply numerators together.
• Multiply denominators together.
• Simplify if possible.

Numerator and Denominator

Understanding the terms 'numerator' and 'denominator' is crucial when working with fractions. In a fraction, the numerator is the top number, which represents the number of parts we have. The denominator is the bottom number, which indicates the total number of equal parts something is divided into.
For example, in the fraction \( \frac{3}{8} \):

• The numerator is \( 3 \), meaning we have 3 parts.
• The denominator is \( 8 \), indicating the whole is divided into 8 equal parts.

When you multiply fractions, you multiply the numerators together and then the denominators. Using the given exercise, multiplying \( -\frac{1}{4} \) and \( -\frac{5}{6} \) involves:

• Numerators: \( -1 \times -5 = 5 \)
• Denominators: \( 4 \times 6 = 24 \)

This results in the fraction \( \frac{5}{24} \).

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This is done by dividing the numerator and the denominator by their greatest common factor (GCF). A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1.
For the fraction \( \frac{5}{24} \):

• The numerator is \( 5 \).
• The denominator is \( 24 \).

You check if there’s a common factor. Since the only common factor of 5 and 24 is 1, \( \frac{5}{24} \) is already in its simplest form.
So, always remember:

• Identify the GCF of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCF.
• Check that the fraction is simplified as much as possible.

This helps ensure your answers are precise and correct.