45 54 simplified

Breiz Heurope

2 min read

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10-03-2025

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Mastering the 45/54 Simplified Fraction: A Comprehensive Guide

The fraction 45/54 might seem daunting at first, but simplifying it is easier than you think. This guide will walk you through the process step-by-step, explaining the underlying math and offering helpful tips for similar fraction simplification problems. We'll explore multiple methods to ensure you understand the concept fully and can tackle any fraction reduction with confidence.

Understanding Fraction Simplification

Before diving into the simplification of 45/54, let's review the basic concept. Simplifying a fraction means reducing it to its lowest terms. This means finding the greatest common divisor (GCD) of the numerator (45) and the denominator (54) and dividing both by that number. The resulting fraction will have the same value as the original but will be expressed in a more concise and manageable form.

Method 1: Finding the Greatest Common Divisor (GCD)

The most straightforward method involves finding the GCD of 45 and 54. Here's how:

1. List the factors: Find all the numbers that divide evenly into 45 and 54.

   • Factors of 45: 1, 3, 5, 9, 15, 45
   • Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54

2. Identify the greatest common factor: The largest number that appears in both lists is 9. Therefore, the GCD of 45 and 54 is 9.

3. Simplify the fraction: Divide both the numerator and the denominator by the GCD (9):

   45 ÷ 9 = 5 54 ÷ 9 = 6

   Therefore, 45/54 simplified is 5/6.

Method 2: Prime Factorization

This method is particularly useful for larger numbers. It involves breaking down the numerator and denominator into their prime factors.

1. Find the prime factorization of 45: 45 = 3 x 3 x 5 = 3² x 5

2. Find the prime factorization of 54: 54 = 2 x 3 x 3 x 3 = 2 x 3³

3. Identify common factors: Both 45 and 54 share two factors of 3 (3²).

4. Simplify: Divide both the numerator and denominator by the common factors (3 x 3 = 9):

   (3² x 5) / (2 x 3³) = 5/ (2 x 3) = 5/6

   Again, the simplified fraction is 5/6.

Method 3: Stepwise Division

If you don't immediately see the GCD, you can simplify the fraction step-by-step by dividing both the numerator and denominator by any common factor you find. You repeat this process until no more common factors exist.

For example:

1. Notice that both 45 and 54 are divisible by 3: 45/3 = 15 and 54/3 = 18. This gives us 15/18.

2. Notice that both 15 and 18 are divisible by 3: 15/3 = 5 and 18/3 = 6. This gives us 5/6.

Since 5 and 6 have no common factors other than 1, we've reached the simplest form: 5/6.

Why Simplify Fractions?

Simplifying fractions makes them easier to understand and work with. A simplified fraction is more concise and easier to visualize. It also simplifies calculations involving fractions, leading to more accurate and efficient results.

Practice Problems

Try simplifying these fractions using the methods described above:

• 12/18
• 24/36
• 35/49

By mastering fraction simplification, you'll build a strong foundation in mathematics, improving your problem-solving skills and confidence in tackling more complex mathematical concepts. Remember, practice makes perfect!