Problem 19 - Algebra A Combined Function

Question

Simplify each fraction. See Examples 8 through \(10 .\) $$ \frac{10}{15} $$

Step-by-Step Solution

Answer

\(\frac{10}{15}\) simplifies to \(\frac{2}{3}\).

Step 1: Identify the Greatest Common Divisor (GCD)

Determine the greatest common divisor of the numerator and denominator of the fraction \( \frac{10}{15} \). The factors of 10 are 1, 2, 5, and 10. The factors of 15 are 1, 3, 5, and 15. The greatest common factor between these two numbers is 5.

Step 2: Divide by GCD

Divide the numerator and the denominator of the fraction by the greatest common divisor (5). This simplifies \( \frac{10}{15} \) as follows:\[ \frac{10 \div 5}{15 \div 5} = \frac{2}{3} \]

Step 3: Verify the Simplified Fraction

Ensure that \( \frac{2}{3} \) is in its simplest form by checking that the numerator and denominator have no common factors other than 1. The factors of 2 are 1 and 2, and the factors of 3 are 1 and 3. Both numbers have no common factors, so \( \frac{2}{3} \) is indeed simplified.

Key Concepts

Greatest Common Divisor

The Greatest Common Divisor, often abbreviated as GCD, is a key concept in simplifying fractions. When we simplify a fraction, we look for the largest number that can divide both the numerator and the denominator evenly. This number is called the GCD.

By finding the GCD, we can reduce the fraction to its simplest form by dividing the numerator and the denominator by this number. For instance, in the fraction \( \frac{10}{15} \), the factors of 10 are 1, 2, 5, and 10, while the factors of 15 are 1, 3, 5, and 15. The largest number in both lists of factors is 5, thus the GCD is 5.

To simplify the fraction, divide both 10 and 15 by 5, leading to \( \frac{2}{3} \). This way, the GCD helps ensure that the fraction is as simple as possible.

Numerator and Denominator

Every fraction is made up of two main components: the numerator, which is the top number, and the denominator, which is the bottom number. The numerator tells us how many parts we have, while the denominator tells us how many equal parts make up a whole.

In the fraction \( \frac{10}{15} \), 10 is the numerator and 15 is the denominator. Understanding these two parts is crucial because when simplifying fractions, we need to look for common factors between them. These common factors are what will allow us to reduce the fraction.

• The numerator (10) shows how many parts are considered.
• The denominator (15) shows the total number of equal parts.

By focusing on the relationship between the numerator and denominator, we can simplify fractions effectively.

Simplified Form

A fraction is in its simplified form when there are no common factors other than 1 between the numerator and denominator. Simplifying a fraction means making it easier to read and use, without changing its value.

For example, after finding the GCD of \( \frac{10}{15} \) to be 5, we divide both the numerator and denominator by this GCD. This gives us \( \frac{2}{3} \). To verify that this fraction is indeed in its simplest form, we check the factors:

• The factors of 2 are 1 and 2.
• The factors of 3 are 1 and 3.

Since there are no common factors between 2 and 3 except for 1, \( \frac{2}{3} \) is the simplest form of the fraction \( \frac{10}{15} \). Simplified fractions are easier to work with, especially in calculations such as adding, subtracting, or comparing fractions.