Problem 89 - Algebra 1: Concepts and Skills

Question

Write the fraction in simplest form. (Skills Review p. 763) $$ \frac{5}{15} $$

Step-by-Step Solution

Answer

The simplest form of \( \frac{5}{15} \) is \( \frac{1}{3} \).

Step 1: Identify the Greatest Common Divisor (GCD)

Find the GCD of the numerator and the denominator. In this case, the GCD of 5 and 15 is 5. Since the GCD is not 1, the fraction can be simplified.

Step 2: Divide by GCD

Divide both the numerator and the denominator by the GCD. Thus, \( \frac{5}{15} \) becomes \( \frac{5 \div 5}{15 \div 5} \).

Step 3: Simplify the fraction

After dividing, the fraction becomes \( \frac{1}{3} \). This is the simplest form of the fraction.

Key Concepts

Understanding the Greatest Common Divisor (GCD)

The greatest common divisor, often abbreviated as GCD, is a crucial concept when it comes to simplifying fractions. It is the largest number that can evenly divide two or more numbers. For a fraction, we look at its numerator and denominator to determine their GCD. To find the GCD, list out all the factors of the numerator and the denominator. In our example, the fraction \( \frac{5}{15} \), the factors of 5 are 1 and 5, and the factors of 15 are 1, 3, 5, and 15. The largest number common to both lists is 5, making it the GCD. When the GCD is not 1, you can reduce the fraction. This process ensures the fraction is expressed in the smallest possible denominator and numerator while still being equivalent to the original fraction.

Explaining Numerator and Denominator

In any fraction, you'll find two critical components: the numerator and the denominator. These terms help us understand what a fraction represents.

• The numerator is the top number in a fraction. It shows how many parts we have.
• The denominator is the bottom number. This number tells us into how many parts the whole is divided.

For example, in the fraction \( \frac{5}{15} \), 5 is the numerator, meaning we have 5 parts. The denominator is 15, indicating the whole is divided into 15 parts. By understanding these terms, you can approach simplifying by knowing you're trying to find the largest common measure to reduce both numbers to something smaller but still proportionally equal.

Achieving the Simplest Form

Simplifying a fraction means transforming it into its simplest form, which is the version with the smallest whole number numerator and denominator, while retaining its value. To simplify, divide both the numerator and the denominator by their greatest common divisor. From our steps, we had \( \frac{5}{15} \) with a GCD of 5. Dividing both numbers by 5, we obtain \( \frac{1}{3} \). This new fraction, \( \frac{1}{3} \), has a numerator of 1 and a denominator of 3, neither of which can be divided by a larger common divisor than 1. Thus, there's no further reduction possible. Achieving the simplest form is often necessary for various mathematical applications and provides a clearer understanding of the fraction's proportional representation. It simplifies the process of comparing fractions and performing arithmetic operations on them.