75 90 In Simplest Form

Simplifying Fractions: Understanding 75/90 in its Simplest Form

Finding the simplest form of a fraction is a fundamental concept in mathematics. It's about representing a fraction in its most concise and efficient way, without changing its value. We'll cover the underlying mathematical principles and provide practical examples to solidify your understanding. This article will guide you through the process of simplifying 75/90, explaining the method in detail and addressing common questions. By the end, you'll not only know the simplest form of 75/90 but also be equipped to simplify other fractions with confidence.

Introduction: What is a Fraction?

Before we dive into simplifying 75/90, let's refresh our understanding of fractions. In practice, a fraction represents a part of a whole. That's why it consists of two numbers: the numerator (the top number) and the denominator (the bottom number). The numerator indicates how many parts we have, while the denominator indicates how many equal parts the whole is divided into. As an example, in the fraction 3/4, the numerator is 3 and the denominator is 4. This means we have 3 out of 4 equal parts.

Simplifying Fractions: The Greatest Common Divisor (GCD)

Simplifying a fraction means reducing it to its lowest terms. In practice, this is achieved by finding the greatest common divisor (GCD) of both the numerator and the denominator and then dividing both numbers by the GCD. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.

Several methods can be used to find the GCD. Let's explore two common approaches:

1. Listing Factors:

This method involves listing all the factors (numbers that divide evenly) of both the numerator and the denominator. Then, we identify the largest factor that is common to both lists.

Factors of 75: 1, 3, 5, 15, 25, 75
Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Comparing the lists, we find that the greatest common factor is 15.

2. Prime Factorization:

This method involves breaking down both the numerator and the denominator into their prime factors (numbers divisible only by 1 and themselves). Then, we identify the common prime factors and multiply them together to find the GCD.

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 together (3 x 5 = 15) gives us the GCD, which is 15.

Simplifying 75/90: Step-by-Step

Now that we know the GCD of 75 and 90 is 15, we can simplify the fraction:

Divide the numerator by the GCD: 75 ÷ 15 = 5
Divide the denominator by the GCD: 90 ÷ 15 = 6

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

Visual Representation

Imagine a pizza cut into 90 equal slices. If we group the slices into sets of 15, we'll have 5 sets of 15 slices out of a total of 6 sets of 15 slices. 75/90 represents having 75 of those slices. This visually confirms that 75/90 simplifies to 5/6.

Understanding Equivalent Fractions

Simplifying a fraction doesn't change its value; it just expresses it in a more concise form. So they represent the same proportion or part of a whole. You can verify this by dividing 75 by 90 and 5 by 6; both will give you the same decimal value (0.On the flip side, 75/90 and 5/6 are equivalent fractions. 8333...).

Further Examples: Applying the Simplification Process

Let's practice simplifying other fractions using the methods described above:

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Simplify 12/18:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- GCD = 6
- Simplified fraction: 12 ÷ 6 / 18 ÷ 6 = 2/3
Simplify 24/36:
- Prime factorization of 24: 2³ x 3
- Prime factorization of 36: 2² x 3²
- GCD = 2² x 3 = 12
- Simplified fraction: 24 ÷ 12 / 36 ÷ 12 = 2/3
Simplify 42/56:
- Prime factorization of 42: 2 x 3 x 7
- Prime factorization of 56: 2³ x 7
- GCD = 2 x 7 = 14
- Simplified fraction: 42 ÷ 14 / 56 ÷ 14 = 3/4

Frequently Asked Questions (FAQ)

Q1: What happens if the numerator and denominator have no common factors other than 1?

A1: If the GCD is 1, the fraction is already in its simplest form. It cannot be simplified further.

Q2: Is there a shortcut for simplifying fractions?

A2: While the GCD method is thorough, you can sometimes simplify by dividing both the numerator and denominator by small common factors repeatedly until you reach the simplest form. Which means for example, with 75/90, you could divide both by 5 to get 15/18, and then divide both by 3 to get 5/6. Even so, using the GCD method ensures you reach the simplest form in one step.

Q3: Why is simplifying fractions important?

A3: Simplifying fractions makes them easier to understand and work with. It improves clarity and makes calculations simpler, particularly when dealing with larger numbers or more complex mathematical operations.

Q4: Can I simplify fractions with decimals?

A4: No, the simplification process applies to fractions with whole numbers. To simplify fractions with decimals, first convert them to fractions with whole numbers, then apply the simplification techniques. As an example, 0.75/0.90 can be rewritten as 75/90, and then simplified to 5/6.

Q5: What if I get a negative fraction?

A5: Simplify the fraction as you normally would, ignoring the negative sign. Then, apply the negative sign to the simplified fraction. Here's one way to look at it: -75/90 simplifies to -5/6.

Conclusion: Mastering Fraction Simplification

Simplifying fractions is a crucial skill in mathematics. Understanding the concept of the greatest common divisor (GCD) and the methods for finding it are key to mastering this skill. Whether you use the factor listing method or prime factorization, the goal is to find the largest number that divides both the numerator and the denominator without leaving a remainder. Plus, by consistently applying these techniques, you'll develop confidence in simplifying fractions and improve your overall mathematical abilities. Remember, practice is key! The more you work with fractions, the easier and more intuitive the process will become. The example of 75/90, simplified to 5/6, serves as a clear illustration of this fundamental mathematical process. Now you're equipped to confidently simplify any fraction you encounter!