Problem 5 - Elementary Algebra

Question

Reduce each fraction to lowest terms. $$\frac{15}{9}$$

Step-by-Step Solution

Answer

15/9 simplifies to 5/3.

Step 1: Identify the GCF (Greatest Common Factor)

To reduce the fraction \( \frac{15}{9} \), we first need to find the greatest common factor (GCF) of the numerator and the denominator. List the factors of 15: 1, 3, 5, 15; and the factors of 9: 1, 3, 9. The common factors are 1 and 3, with the greatest being 3.

Step 2: Divide by the GCF

Once the GCF is identified as 3, divide both the numerator and the denominator of \( \frac{15}{9} \) by this number. This means calculating \( \frac{15 \div 3}{9 \div 3} \), resulting in \( \frac{5}{3} \).

Step 3: Verify the Result

Check if \( \frac{5}{3} \) can be reduced further by looking for any common factors between 5 and 3. Since the only common factor is 1, \( \frac{5}{3} \) is in its simplest form.

Key Concepts

Greatest Common Factor

Understanding the concept of the Greatest Common Factor (GCF) is essential when reducing fractions. The GCF of two numbers is the largest number that divides both numbers without leaving a remainder.
To find the GCF:

• List the factors of each number. Factors are numbers you multiply together to get another number.
• Identify which factors are common between the two numbers.
• The greatest or largest of these common factors is the GCF.

For example, to reduce the fraction \( \frac{15}{9} \), we list the factors of the numerator (15) and the denominator (9). The factors of 15 are 1, 3, 5, and 15. The factors of 9 are 1, 3, and 9.

The common factors are 1 and 3, and the GCF is 3. By dividing both the numerator and denominator by this GCF, we effectively reduce the fraction to its simplest form.

Numerator and Denominator

In a fraction, the number on the top is called the numerator, and the number on the bottom is the denominator. Understanding these terms is crucial in working with fractions.
The numerator represents the number of parts we have, while the denominator shows the total number of equal parts in a whole.

For example, in the fraction \( \frac{15}{9} \):

• 15 is the numerator, indicating we have 15 parts.
• 9 is the denominator, indicating that the whole is divided into 9 equal parts.

When reducing fractions, both the numerator and denominator are divided by their GCF. This process ensures that the fraction maintains the same value, even though the numbers are smaller. This simplification results in the most reduced form of the fraction without changing its value.

Simplifying Fractions

Simplifying fractions makes them easier to understand and work with, especially in calculations. To simplify a fraction means to reduce it to its simplest form, where the numerator and denominator have no common factors other than 1.
The process involves:

• Finding the GCF of the numerator and denominator.
• Dividing both the numerator and denominator by this GCF.
• Checking the result to ensure there are no further common factors.

Let's use the fraction \( \frac{15}{9} \) as an example. To simplify:

• Identify the GCF, which we found to be 3.
• Divide both the numerator (15) and the denominator (9) by 3, resulting in \( \frac{5}{3} \).
• Since 5 and 3 have no common factors other than 1, \( \frac{5}{3} \) is simplified completely.

This simplified fraction represents the same proportion as the original but is presented in its cleanest form, making it more straightforward for further mathematical operations.