45 54 simplified

Daniel Parker

2 min read

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01-03-2025

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Simplifying Fractions: A Deep Dive into 45/54

Understanding how to simplify fractions is a fundamental skill in mathematics. This article will guide you through the process of simplifying the fraction 45/54, explaining the underlying concepts and providing a step-by-step approach. We'll also explore how to apply this method to other fractions.

What Does Simplifying a Fraction Mean?

Simplifying a fraction, also known as reducing a fraction to its lowest terms, means finding an equivalent fraction where the numerator (top number) and denominator (bottom number) have no common factors other than 1. In essence, we're expressing the fraction in its most concise form.

Step-by-Step Simplification of 45/54

To simplify 45/54, we need to find the greatest common divisor (GCD) of 45 and 54. The GCD is the largest number that divides both 45 and 54 without leaving a remainder.

1. Find the factors of 45: 1, 3, 5, 9, 15, 45
2. Find the factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
3. Identify the common factors: 1, 3, 9
4. Determine the greatest common factor (GCF): The largest common factor is 9.

Now, we divide both the numerator and the denominator by the GCD (9):

45 ÷ 9 = 5 54 ÷ 9 = 6

Therefore, the simplified fraction is 5/6.

Understanding the Process: Why it Works

Dividing both the numerator and the denominator by their GCD doesn't change the value of the fraction. It's like dividing a pizza into 45 slices and taking 9 of them, or dividing the same pizza into 54 slices and taking 18. Both scenarios represent the same portion of the whole pizza. Simplifying merely changes the way we represent that proportion.

Methods for Finding the GCD

While listing factors works well for smaller numbers, it becomes less efficient with larger ones. Here are alternative methods:

• Prime Factorization: Break down both numbers into their prime factors. The GCD is the product of the common prime factors raised to the lowest power.

  • 45 = 3² x 5
  • 54 = 2 x 3³
  • Common prime factor: 3² = 9. Therefore, the GCD is 9.

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

Simplifying Other Fractions: A Practical Approach

The same principles apply to simplifying any fraction. Always look for the greatest common divisor and divide both the numerator and the denominator by it. Practice will help you quickly recognize common factors and simplify fractions efficiently.

Examples:

• 12/18 (GCD = 6) simplifies to 2/3
• 24/36 (GCD = 12) simplifies to 2/3
• 15/25 (GCD = 5) simplifies to 3/5

Mastering fraction simplification is crucial for further mathematical concepts, particularly in algebra and beyond. By understanding the underlying principles and practicing different methods, you'll build a solid foundation for future success in mathematics.