Problem 22 - Algebra A Combined Function

Question

Simplify each fraction. See Examples 8 through \(10 .\) $$ \frac{5}{9} $$

Step-by-Step Solution

Answer

The fraction \( \frac{5}{9} \) is already simplified.

Step 1: Understand the Fractions

The fraction given is \( \frac{5}{9} \). We need to determine if any common factors exist between the numerator (5) and the denominator (9) that can be used to simplify the fraction.

Step 2: Find Common Factors

Identify any common factors between the numbers 5 and 9. Since 5 is a prime number, it can only be divided by 1 and itself. The greatest common factor between 5 and 9 is 1.

Step 3: Simplify the Fraction

Since the greatest common factor is 1, divide both the numerator and the denominator by 1. Therefore, the fraction \( \frac{5}{9} \) remains the same because dividing by 1 does not change the value of the numbers.

Key Concepts

Common Factors

When simplifying fractions, one crucial step is identifying the common factors of the numerator and denominator. Common factors are numbers that divide each of these numbers without a remainder. For instance, consider the fraction \(\frac{12}{16}\).

• List the factors of the numerator (12): 1, 2, 3, 4, 6, 12.
• List the factors of the denominator (16): 1, 2, 4, 8, 16.

The numbers they both share are 1, 2, and 4, making these the common factors. Finding these is essential because they guide us towards simplifying the fraction by showing what numbers we can divide both parts of the fraction by, to reduce it in a balanced way.

Prime Numbers

Understanding prime numbers helps in simplifying fractions, especially when dealing with small numbers. A prime number is a natural number greater than 1 that has no divisors other than 1 and itself.
Prime numbers include 2, 3, 5, 7, 11 and so on. If either the numerator or the denominator of a fraction is a prime number, it limits the common factors to 1 and perhaps the prime number itself if it appears in both parts of the fraction.
For the fraction \(\frac{5}{9}\):

• The number 5 is prime, so its only factors are 1 and 5.
• The number 9 isn't prime because it can be divided by 1, 3, and 9.

Since 5 and 9 have no common factors other than 1, checking for prime numbers can quickly indicate when a fraction is already as simple as it can be.

Greatest Common Factor

The Greatest Common Factor (GCF) is the highest factor that two numbers share. It's a pivotal concept when simplifying fractions because it signifies the maximum number by which we can divide both the numerator and the denominator without leaving a remainder.
Finding the GCF involves:

• Identifying the factors of each number.
• Highlighting the common factors.
• Selecting the largest common factor.

In the case of \(\frac{5}{9}\):

• The factors of 5 are: 1, 5.
• The factors of 9 are: 1, 3, 9.

The only common factor is 1. Thus, the GCF is 1, indicating that \(\frac{5}{9}\) is already in its simplest form. Understanding the GCF is crucial because it ensures we don't oversimplify or compromise the integrity of the fraction's value.