75 90 In Simplest Form

Simplifying Fractions: Understanding 75/90

The seemingly simple fraction 75/90 might look intimidating at first glance, but understanding how to simplify it is a fundamental skill in mathematics. This article will guide you through the process of simplifying 75/90 to its simplest form, explaining the concept in a clear, step-by-step manner. We'll cover the underlying principles, explore different methods, and answer frequently asked questions to ensure a complete understanding. This comprehensive guide will help you confidently tackle similar fraction simplification problems.

Understanding Fraction Simplification

Before diving into the specific example of 75/90, let's establish a foundational understanding of what fraction simplification means. A fraction represents a part of a whole. It's expressed as a numerator (the top number) over a denominator (the bottom number). Simplifying a fraction means reducing it to its lowest terms, meaning the numerator and denominator have no common factors other than 1. This doesn't change the value of the fraction; it simply represents it in a more concise and manageable form. Think of it like reducing a recipe; you can halve the ingredients, but the final dish will taste the same.

Finding the Greatest Common Divisor (GCD)

The key to simplifying any fraction lies in finding the greatest common divisor (GCD), also known as the greatest common factor (GCF), of the numerator and the denominator. The GCD is the largest number that divides both the numerator and denominator without leaving a remainder. There are several ways to find the GCD:

• Listing Factors: Write down all the factors of both numbers and identify the largest one they share. For example, the factors of 75 are 1, 3, 5, 15, 25, and 75. The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90. The largest factor they share is 15.

• Prime Factorization: This method involves breaking down each number into its prime factors (numbers divisible only by 1 and themselves). Then, identify the common prime factors and multiply them together.

  • Prime factorization of 75: 3 x 5 x 5 = 3 x 5²
  • Prime factorization of 90: 2 x 3 x 3 x 5 = 2 x 3² x 5

  The common prime factors are 3 and 5. Multiplying them gives us 15, which is the GCD.

• Euclidean Algorithm: This is a more efficient method for larger numbers. It involves repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until the remainder is 0. The last non-zero remainder is the GCD. Let's illustrate this with 75 and 90:

  1. 90 ÷ 75 = 1 with a remainder of 15
  2. 75 ÷ 15 = 5 with a remainder of 0

  The last non-zero remainder is 15, so the GCD is 15.

Simplifying 75/90: A Step-by-Step Guide

Now that we understand how to find the GCD, let's apply it to simplify 75/90:

1. Find the GCD: As we determined above, the GCD of 75 and 90 is 15.

2. Divide both the numerator and the denominator by the GCD:

   • 75 ÷ 15 = 5
   • 90 ÷ 15 = 6

3. Write the simplified fraction: The simplified fraction is 5/6.

Therefore, 75/90 simplified to its lowest terms is 5/6.

Visualizing Fraction Simplification

Imagine you have a pizza cut into 90 slices. The fraction 75/90 represents having 75 of those 90 slices. Simplifying the fraction means grouping those slices. If we group the slices into sets of 15, we'll have 5 groups of 15 slices out of a total of 6 groups of 15 slices. This visually represents the simplified fraction 5/6.

Further Applications and Extensions

The process of simplifying fractions is crucial in various mathematical contexts, including:

• Algebra: Simplifying fractions is essential for manipulating algebraic expressions and solving equations.

• Geometry: Fractions are commonly used in geometric calculations, such as finding areas and volumes. Simplifying fractions makes these calculations more efficient and easier to understand.

• Real-World Applications: Fractions are used extensively in everyday life, from cooking and baking to calculating proportions and percentages. Simplifying fractions helps us to make quick and accurate calculations.

Frequently Asked Questions (FAQ)

• What if the GCD is 1? If the GCD of the numerator and denominator is 1, the fraction is already in its simplest form. It cannot be simplified further.

• Can I simplify a fraction by dividing the numerator and denominator by different numbers? No, you must divide both the numerator and the denominator by the same number (the GCD) to maintain the value of the fraction.

• Are there any shortcuts for simplifying fractions? While the methods described above are generally the most reliable, recognizing common factors can sometimes speed up the process. For example, if both the numerator and denominator are even numbers, you can immediately divide both by 2.

• What if the numerator is larger than the denominator? This is called an improper fraction. You can simplify it using the same methods as with proper fractions. You can also convert it into a mixed number (a whole number and a fraction).

Conclusion

Simplifying fractions, as demonstrated through the example of 75/90, is a fundamental mathematical operation with broad applications. Mastering this skill is essential for success in various mathematical fields and real-world scenarios. By understanding the concept of the greatest common divisor (GCD) and employing the appropriate methods, you can confidently simplify any fraction to its lowest terms. Remember, practice is key to improving your proficiency in simplifying fractions. Start with simpler examples, gradually increasing the complexity, and soon you'll be simplifying fractions with ease. The journey to mathematical fluency starts with understanding fundamental concepts like this one, so keep practicing and keep exploring!