Problem 24 - Prealgebra

Question

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. (lesson \(4-4\) ) $$\frac{6 r}{15 r s}$$

Step-by-Step Solution

Answer

The fraction simplifies to \(\frac{2}{5s}\).

Step 1: Identify Common Factors

The first step is to identify the common factors in the numerator and the denominator. The numerator is \(6r\) and the denominator is \(15rs\). The common factor here is \(3r\) because both \(6r\) and \(15rs\) are divisible by \(3r\).

Step 2: Simplify the Fraction

Divide both the numerator and the denominator by their greatest common factor, \(3r\). This gives:\[\frac{6r \div 3r}{15rs \div 3r} = \frac{2}{5s}\]

Step 3: Confirm Simplification

Finally, check if the simplified fraction \(\frac{2}{5s}\) can be reduced further. Since \(2\) and \(5s\) have no common factors, \(\frac{2}{5s}\) is in its simplest form.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is the largest number that can evenly divide both the numerator and the denominator of a fraction. When simplifying fractions, finding the GCF is crucial because it allows us to reduce the fraction to its simplest form. For instance, in the fraction \(\frac{6r}{15rs}\), we need to identify the largest algebraic factor that divides both parts.

To do so, you can follow these steps:

• List the factors of both the numerator and denominator.
• Identify the common factors between them.
• Choose the largest common factor as the GCF.

In our example, both \(6r\) (numerator) and \(15rs\) (denominator) have a common factor \(3r\). This is because \(6r\) and \(15rs\) can both be divided evenly by \(3r\), making it the greatest common factor. Finding this GCF is the first step towards simplifying a fraction.

Simplest Form

When a fraction is expressed in its simplest form, it means no common factor other than 1 exists between the numerator and the denominator. Simplifying ensures the fraction is as reduced as possible, making it easier to understand and work with.

To simplify a fraction:

• First, determine the GCF of the numerator and denominator.
• Divide both the numerator and the denominator by this GCF.

In our example, using the GCF \(3r\), the fraction \(\frac{6r}{15rs}\) simplifies to \(\frac{2}{5s}\). Since there are no further common factors between 2 and \(5s\), the fraction is now in its simplest form. Reducing fractions to their simplest form is essential in math, as it improves clarity and avoids overly complex expressions.

Numerator and Denominator

A fraction consists of two main parts: the numerator and the denominator. The numerator is the top number that represents how many parts you have, while the denominator, the bottom number, indicates into how many parts the whole is divided.

In the fraction \(\frac{6r}{15rs}\):

• \(6r\) is the numerator.
• \(15rs\) is the denominator.

Understanding the roles of these parts is pivotal for simplifying fractions. The numerator and denominator must be looked at together to discern the greatest common factor. This approach helps simplify the fraction, as seen in our example, where dividing both by \(3r\) led us to the simpler form \(\frac{2}{5s}\). Grasping the concept of numerator and denominator is a vital step in becoming adept with fractions.