Fractions are an essential aspect of mathematics and play a crucial role in various real-life situations. However, for many people, even adults, the concept of fractions can be intimidating and confusing. If you find yourself struggling with fractions, fear not! This step-by-step guide will walk you through the process of simplifying fractions, making it easy and accessible for adults who need help understanding them.

Let's get started!

Before we delve into the process of simplification, let's ensure we have a solid grasp of what fractions are:

  • A fraction represents a part of a whole and is written in the form of a numerator (top number) over a denominator (bottom number).
  • The numerator indicates the number of parts we have, while the denominator represents the total number of equal parts that make up the whole.
Woman sitting in front of a whiteboard with math problems.

Simplifying fractions is the process of reducing a fraction to its most straightforward and smallest form. When a fraction is simplified, the numerator and denominator have no common factors other than 1.

To simplify fractions, we need to find the Greatest Common Divisor (GCD) (also known as the Highest Common Factor or HCF) of the numerator and the denominator. The GCD is the largest number that divides both numbers evenly. Here's how to do it:

Method 1 - Prime Factorization

  1. List the prime factors of both the numerator and the denominator separately.
  2. Identify the common prime factors shared by both numbers.
  3. Multiply the common prime factors to find the GCD.

Example: Simplify 24/36

Step 1: Prime factors of 24: 2 × 2 × 2 × 3 Prime factors of 36: 2 × 2 × 3 × 3

Step 2: Common prime factors: 2 × 2 × 3 = 12

Step 3: The GCD of 24 and 36 is 12.

Method 2 - Euclidean Algorithm

The Euclidean Algorithm is a more efficient method, particularly for larger numbers. Follow these steps:

  1. Divide the larger number by the smaller number.
  2. Replace the larger number with the remainder from the division.
  3. Repeat the process until the remainder is 0.
  4. The last non-zero remainder is the GCD.

Example: Find the GCD of 48 and 18 using the Euclidean Algorithm.

Step 1: 48 ÷ 18 = 2 with a remainder of 12

Step 2: 18 ÷ 12 = 1 with a remainder of 6

Step 3: 12 ÷ 6 = 2 with a remainder of 0

The GCD of 48 and 18 is 6.

Now that we have the GCD, we can proceed to simplify the fraction using the following steps:

  1. Divide both the numerator and the denominator by the GCD to get the simplified fraction.
  2. If the result after division is a whole number, it can be written as a fraction with the whole number over 1.
  3. For improper fractions (numerator greater than or equal to the denominator), convert them to mixed numbers.

Example 1 - Simplify 36/12

Step 1: GCD of 36 and 12 = 12

Step 2: Divide both the numerator and the denominator by 12:

36 ÷ 12 = 3 12 ÷ 12 = 1

Step 3: The result is 3, which can be expressed as the mixed number 3/1 or simply 3.

Example 2 - Simplify 14/7

Step 1: GCD of 14 and 7 = 7

Step 2: Divide both the numerator and the denominator by 7:

14 ÷ 7 = 2 7 ÷ 7 = 1

Step 3: The result is 2, which can be expressed as the mixed number 2/1 or simply 2.

In some cases, you may encounter complex fractions where the numerator and/or denominator are themselves fractions. The process of simplification remains the same:

  1. Simplify the numerator and denominator separately using the steps described above.
  2. Continue to simplify until you have a fully reduced fraction.

Example - Simplify (6/8)/(15/20)

Step 1: Simplify the numerator and denominator separately: 6/8 simplifies to 3/4 15/20 simplifies to 3/4

Step 2: Now, simplify the entire complex fraction:

(3/4)/(3/4) = 1

The result is 1.

Once you've simplified a fraction, it's beneficial to know its decimal and percentage equivalents. Here's how you can do it:

  1. To convert a fraction to a decimal, divide the numerator by the denominator.
  2. To convert a decimal to a percentage, multiply the decimal by 100.

Example - Convert 3/5 to a Decimal and Percentage

Step 1: 3 ÷ 5 = 0.6

Step 2: 0.6 × 100 = 60%

Understanding and simplifying fractions doesn't have to be a daunting task. By following this step-by-step guide, you can gain confidence in handling fractions, making them less intimidating and more approachable. Remember, practice makes perfect, so keep honing your skills, and soon you'll be a fraction-simplifying pro! Happy fraction-simplifying!