Problem 13 - Beginning and Intermediate Algebra

Question

Write the fraction in lowest terms. $$\frac{10}{15}$$

Step-by-Step Solution

Answer

\(\frac{2}{3}\)

Step 1: Identify the GCD

To reduce a fraction to its lowest terms, start by finding the greatest common divisor (GCD) of the numerator and the denominator. Here, we need to find the GCD of 10 and 15.

Step 2: List Factors

List the factors of each number. Factors of 10 are 1, 2, 5, and 10. Factors of 15 are 1, 3, 5, and 15. Identify the greatest common factor.

Step 3: Determine the GCD

The greatest common factor from the list is 5. Thus, the GCD of 10 and 15 is 5.

Step 4: Divide by the GCD

Divide both the numerator and the denominator of the fraction by the GCD to simplify it. Divide 10 by 5 to get 2, and divide 15 by 5 to get 3.

Step 5: Write the Simplified Fraction

The result from dividing both the numerator and denominator by the GCD gives us the fraction in its simplest form: \(\frac{2}{3}\).

Key Concepts

Greatest Common Divisor

Understanding how to simplify fractions starts with knowing about the greatest common divisor, or GCD. The GCD is the largest number that can exactly divide both the numerator and the denominator of a fraction without leaving a remainder. Finding the GCD is crucial because it helps in reducing the fraction to its simplest form.

• First, list out the factors of the numerator and the denominator.
• Check which number is the largest common factor in both lists.

For example, if we take the fraction \( \frac{10}{15} \), we list the factors of 10 (1, 2, 5, 10) and the factors of 15 (1, 3, 5, 15). Here, the greatest common factor is 5, making it the GCD for both numbers.

Numerator and Denominator

To simplify a fraction, we must understand the roles of the numerator and the denominator. In a fraction \( \frac{a}{b} \), "a" is the numerator and "b" is the denominator. The numerator represents how many parts of a whole you have, and the denominator tells you how many parts make up that whole.

• The numerator is the top number in a fraction.
• The denominator is the bottom number in a fraction.

For our fraction \( \frac{10}{15} \), 10 is how many parts we are considering, while 15 is the total number of equal parts in one whole item. Recognizing these roles is essential as we work through the process of simplifying fractions.

Fractions in Lowest Terms

Fractions in their lowest terms are the simplest form of a fraction. This means the numerator and the denominator can no longer be divided by any common factor other than 1. Achieving this requires dividing both the numerator and the denominator by their greatest common divisor (GCD).

Steps to Simplify

1. Start with the original fraction.
2. Find the GCD of the numerator and the denominator.
3. Divide both numerator and denominator by the GCD.
For our example, \( \frac{10}{15} \), the GCD is 5. Dividing both 10 and 15 by 5 gives you 2 and 3, respectively. Thus, the fraction simplifies to \( \frac{2}{3} \). This is the fraction in its lowest terms, as 2 and 3 have no common factor other than 1.