Problem 17 - Beginning and Intermediate Algebra

Question

Write the fraction in lowest terms. $$\frac{18}{30}$$

Step-by-Step Solution

Answer

\(\frac{3}{5}\)

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify \( \frac{18}{30} \), we first find the greatest common divisor (GCD) of the numerator (18) and the denominator (30). Both numbers can be divided by 2 and 3. The largest number both can be divided by is 6.

Step 2: Divide the Numerator and Denominator by the GCD

Given the GCD is 6, divide the numerator (18) and the denominator (30) by the GCD.\[ \frac{18}{6} = 3 \quad \frac{30}{6} = 5 \]

Step 3: Write the Fraction in Simplest Form

After dividing both the numerator and the denominator by their GCD, we write the fraction in its simplest form:\( \frac{18}{30} = \frac{3}{5} \).

Key Concepts

Greatest Common Divisor

Understanding the greatest common divisor (GCD) is crucial when simplifying fractions. The GCD is the largest positive number that divides both the numerator and the denominator evenly, without leaving a remainder. Finding this number is the first step in reducing fractions to their simplest form.

To find the GCD of two numbers, there are several methods you can use:

• Listing Factors: Write down all factors of the numerator and the denominator, and identify the largest number common to both lists.
• Prime Factorization: Break down both numbers into their prime factors and multiply the common prime factors.
• Euclidean Algorithm: Use division to repeatedly divide the larger number by the smaller number until the remainder is zero. The last non-zero remainder is the GCD.

In our example, the GCD of 18 and 30 is 6, because 6 is the largest number that can divide both 18 and 30 without a remainder.

Numerator and Denominator

Fractions consist of two parts: the numerator and the denominator. It's essential to understand both parts to simplify fractions effectively.

The numerator is the top number in a fraction. It represents the number of parts being considered. For example, in the fraction \( \frac{18}{30} \), 18 is the numerator.

The denominator is the bottom number. It indicates the total number of equal parts into which the whole is divided. In the fraction \( \frac{18}{30} \), 30 is the denominator.

When simplifying a fraction, both the numerator and the denominator must be divided by the same number, in this case, the GCD, to maintain the proportion and value of the fraction.

Simplest Form

A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. This means the fraction can't be reduced further. Simplifying fractions to their simplest form makes them easier to understand and compare.

To achieve the simplest form, follow these steps:

• Calculate the GCD of the numerator and denominator.
• Divide both the numerator and denominator by the GCD.

When applying this to the fraction \( \frac{18}{30} \), we find that the GCD is 6. Dividing both 18 and 30 by 6 gives us \( \frac{3}{5} \), which is the simplest form of the fraction. This process ensures that the fraction is expressed in the smallest numerals possible while maintaining its original value.