Problem 30 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each fraction by reducing it to its lowest terms. $$\frac{8}{14}$$

Step-by-Step Solution

Answer

The simplified form of the fraction \( \frac{8}{14} \) is \( \frac{4}{7} \)

Step 1: Identifying the Greatest Common Divisor (GCD)

First, identify the largest number that evenly divides both 8 and 14. In this case, the greatest common divisor is 2.

Step 2: Dividing Numerator and Denominator by the GCD

Next, divide both the numerator (8) and the denominator (14) by the GCD (2). Doing this gives \( \frac{8}{2} \) and \( \frac{14}{2} \)

Step 3: Simplifying the Fraction

Finally, simplify the fraction by performing the division. This gives \( \frac{8}{2} = 4 \) and \( \frac{14}{2} = 7 \). Therefore, our simplified fraction is \( \frac{4}{7} \)

Key Concepts

Greatest Common Divisor

The greatest common divisor, or GCD, is a key concept when simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator of a fraction. For instance, in the fraction \( \frac{8}{14} \), finding the GCD involves determining the biggest number that divides both 8 and 14 without leaving a remainder.

There are several methods to find the GCD, but a straightforward one is to list the factors of each number. For 8, the factors are 1, 2, 4, and 8. For 14, the factors are 1, 2, 7, and 14. Comparing these factors, the greatest common divisor that both share is 2.

Another common method is using the Euclidean algorithm, a process of repeated division. Despite the method used, once you have the GCD, you can greatly simplify or "reduce" the fraction, making numbers easier to work with or understand.

Numerator and Denominator

Understanding the numerator and the denominator is essential in grasping the concept of fractions. A fraction is made up of these two parts.

• The numerator is the top part of a fraction. It indicates how many parts of a whole are being considered.
• The denominator is the bottom part. It shows the total number of equal parts the whole is divided into.

For example, in the fraction \( \frac{8}{14} \), 8 is the numerator, and 14 is the denominator. This fraction tells us that we are considering 8 parts out of a total of 14. By understanding these components, you can effectively communicate and solve problems involving fractional quantities.

Always remember: changing the numerator or denominator changes the actual value of the fraction. This is why simplifying, which doesn't alter the fraction's value, is very crucial.

Reducing Fractions

Reducing fractions, also known as simplifying, involves expressing a fraction in its lowest terms. This means having the smallest numbers possible in the numerator and denominator while keeping the same value.

To reduce a fraction:

• Find the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by this number.

In our given example, \( \frac{8}{14} \), the GCD is 2. By dividing both 8 and 14 by 2, you simplify the fraction to \( \frac{4}{7} \).

Why is reducing fractions important? It helps in making calculations easier and provides a clearer understanding of relationships between numbers. Simplified fractions often look cleaner and are preferred in both academic and real-world scenarios.