Problem 72 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{6}{14}$$

Step-by-Step Solution

Answer

The reduced form of \( \frac{6}{14} \) is \( \frac{3}{7} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce the fraction \( \frac{6}{14} \) to its lowest terms, we need to find the greatest common divisor of the numerator and the denominator. The factors of 6 are 1, 2, 3, and 6. The factors of 14 are 1, 2, 7, and 14. The common factors are 1 and 2. Thus, the greatest common divisor is 2.

Step 2: Divide the Numerator and Denominator by the GCD

Divide both the numerator and the denominator of the fraction by the GCD, which is 2. This operation is performed as follows: \( \frac{6}{2} = 3 \) and \( \frac{14}{2} = 7 \).

Step 3: Write the Reduced Fraction

Using the results from Step 2, the fraction \( \frac{6}{14} \) can now be expressed in its lowest terms as \( \frac{3}{7} \).

Key Concepts

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is a key concept when working with fractions. It’s the largest number that can evenly divide both the numerator and the denominator of a fraction. Finding the GCD is an important first step in reducing a fraction. For example, in the fraction \( \frac{6}{14} \), we determine that the factors of 6 are \( 1, 2, 3, \) and \( 6 \), and the factors of 14 are \( 1, 2, 7, \) and \( 14 \). The common factors are \( 1 \) and \( 2 \), making \( 2 \) the greatest common divisor. Learning to find the GCD helps make the process of simplifying fractions much simpler. You can do this through listing factors or using the Euclidean algorithm for larger numbers.

Reducing Fractions

Reducing fractions involves simplifying them to their most basic form. This is done by dividing both the numerator and the denominator by their Greatest Common Divisor (GCD). Let's take \( \frac{6}{14} \) as an example. After identifying the GCD as 2, the next step is to divide the numerator and denominator by this number. When you divide \( 6 \) by \( 2 \), the result is \( 3 \). Similarly, dividing \( 14 \) by \( 2 \) results in \( 7 \). Thus, \( \frac{6}{14} \) reduces to \( \frac{3}{7} \). This process maintains the fraction's value while making it simpler to work with.

Numerator and Denominator

In any given fraction, two essential parts are present: the numerator and the denominator. The numerator is the top number, representing how many parts of the whole we have. The denominator, being the bottom number, tells us into how many equal parts the whole is divided. For instance, in the fraction \( \frac{6}{14} \), \( 6 \) is the numerator, and \( 14 \) is the denominator. Understanding these roles helps us perform accurate mathematical operations like addition, subtraction, and especially simplifying fractions.

Lowest Terms

A fraction is said to be in its lowest terms (or simplest form) when the numerator and the denominator have no other common factor besides 1. To achieve this, start by finding the Greatest Common Divisor (GCD) of the numerator and denominator, then divide both by this number. Following through our example of \( \frac{6}{14} \), when simplified by its GCD of 2, the fraction becomes \( \frac{3}{7} \). Since 3 and 7 have no common factors other than 1, \( \frac{3}{7} \) is the lowest term for \( \frac{6}{14} \). Placing a fraction in its lowest terms simplifies further arithmetic operations and makes comparisons easier.