75 90 In Simplest Form

Simplifying Fractions: A Deep Dive into 75/90

Understanding fractions is a fundamental concept in mathematics, crucial for everything from basic arithmetic to advanced calculus. This article will explore the simplification of the fraction 75/90, demonstrating the process in detail and providing a broader understanding of fraction simplification techniques. We'll delve into the underlying mathematical principles, address common questions, and even explore real-world applications. By the end, you'll not only know the simplest form of 75/90 but also possess the skills to simplify any fraction confidently.

What is a Fraction?

Before we tackle 75/90, let's quickly review what a fraction represents. A fraction, like 75/90, is a way of expressing a part of a whole. The top number, 75, is called the numerator, and it represents the number of parts we have. The bottom number, 90, is the denominator, indicating the total number of equal parts the whole is divided into.

Simplifying Fractions: The Basics

Simplifying a fraction means reducing it to its lowest terms. This means finding an equivalent fraction where the numerator and denominator have no common factors other than 1. We achieve this by dividing both the numerator and denominator by their greatest common divisor (GCD).

Finding the Greatest Common Divisor (GCD)

The GCD of two numbers is the largest number that divides both without leaving a remainder. There are several ways to find the GCD:

• Listing Factors: List all the factors of both numbers and find the largest factor 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 common factor is 15.

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

  • Prime factorization of 75: 3 x 5 x 5 (or 3 x 5²)
  • Prime factorization of 90: 2 x 3 x 3 x 5 (or 2 x 3² x 5)
  • Common prime factors: 3 and 5.
  • GCD: 3 x 5 = 15

• Euclidean Algorithm: This is a more efficient method for larger numbers. It involves repeatedly applying the division algorithm until the remainder is 0. The last non-zero remainder is the GCD. Let's demonstrate:

  1. Divide 90 by 75: 90 = 1 x 75 + 15
  2. Divide 75 by the remainder (15): 75 = 5 x 15 + 0 The last non-zero remainder is 15, so the GCD of 75 and 90 is 15.

Simplifying 75/90

Now that we've found the GCD of 75 and 90 to be 15, we can simplify the fraction:

75 ÷ 15 / 90 ÷ 15 = 5/6

Therefore, the simplest form of 75/90 is 5/6.

Visual Representation

Imagine you have a pizza cut into 90 slices. You have 75 of those slices. Simplifying the fraction 75/90 means finding an equivalent representation with fewer slices but maintaining the same proportion. Dividing both the numerator and denominator by 15 is like grouping the pizza slices into sets of 15. You would then have 5 sets of 15 slices out of a total of 6 sets, resulting in the simplified fraction 5/6.

Real-World Applications

Fraction simplification is used extensively in various fields:

• Cooking: Adjusting recipes – If a recipe calls for 75 grams of sugar for 90 cookies, you can simplify the ratio to 5 grams of sugar per 6 cookies.

• Construction: Calculating material quantities – If a project requires 75 bricks for every 90 square feet, the simplified ratio is 5 bricks per 6 square feet.

• Finance: Working with percentages – A discount of 75 out of 90 dollars can be simplified to a discount of 5 out of 6 dollars.

• Data Analysis: Representing proportions – When working with data representing parts of a whole, simplification makes the data easier to understand and interpret.

Further Exploration of Fraction Simplification

The process demonstrated with 75/90 applies to all fractions. To simplify any fraction:

1. Find the GCD of the numerator and denominator. Use any of the methods discussed above (listing factors, prime factorization, or the Euclidean algorithm).

2. Divide both the numerator and denominator by the GCD. This will result in an equivalent fraction in its simplest form.

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. This means the numerator and denominator share no common factors other than 1.

• Can I simplify a fraction by dividing the numerator and denominator by any common factor? Yes, you can, but it's more efficient to divide by the greatest common factor. This ensures that you reach the simplest form in a single step. For example, you could divide 75/90 by 3 to get 25/30, then by 5 to get 5/6. However, dividing by the GCD (15) directly gets you to the answer in one step.

• What if the fraction is a mixed number? Convert the mixed number to an improper fraction first, then simplify using the same method.

• Are there any online tools to simplify fractions? Yes, many online calculators can simplify fractions instantly. However, understanding the underlying process is crucial for building a strong mathematical foundation.

Conclusion

Simplifying fractions, as illustrated with the example of 75/90, is a fundamental skill in mathematics with wide-ranging applications. Mastering this process involves understanding the concept of the greatest common divisor and applying the appropriate method to find it. While online tools are available, understanding the manual process is invaluable for building a strong conceptual understanding and developing problem-solving skills. The ability to simplify fractions not only enhances your mathematical abilities but also aids in interpreting and applying mathematical concepts in various real-world situations. Remember, the simplest form of 75/90 is 5/6, a result achieved by dividing both the numerator and denominator by their greatest common divisor, 15.