45 63 simplified

Daniel Parker

2 min read

·

25-02-2025

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The fraction 45/63 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 concepts and showing you how to arrive at the simplest form of this fraction. We'll also explore the underlying mathematical principles involved.

Understanding Fraction Simplification

Simplifying a fraction, also known as reducing a fraction to its lowest terms, means finding an equivalent fraction where the numerator (top number) and the denominator (bottom number) have no common factors other than 1. This makes the fraction easier to understand and work with.

Finding the Greatest Common Factor (GCF)

The key to simplifying fractions is finding the greatest common factor (GCF) of the numerator and the denominator. The GCF is the largest number that divides both numbers evenly without leaving a remainder.

There are a few ways to find the GCF:

• Listing Factors: List all the factors of each number and identify the largest one they share. For example, the factors of 45 are 1, 3, 5, 9, 15, and 45. The factors of 63 are 1, 3, 7, 9, 21, and 63. The largest factor they share is 9.

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

  • 45 = 3 x 3 x 5 = 3² x 5
  • 63 = 3 x 3 x 7 = 3² x 7

  The common prime factors are 3 and 3 (or 3²). Therefore, the GCF is 3 x 3 = 9.

• Euclidean Algorithm: This method is particularly useful for larger numbers. It involves repeatedly dividing the larger number by the smaller number and taking the remainder until the remainder is 0. The last non-zero remainder is the GCF.

Simplifying 45/63

Now that we know the GCF of 45 and 63 is 9, we can simplify the fraction:

1. Divide both the numerator and the denominator by the GCF:

   45 ÷ 9 = 5 63 ÷ 9 = 7

2. The simplified fraction is: 5/7

Therefore, 45/63 simplified is 5/7.

Why Simplify Fractions?

Simplifying fractions makes them easier to:

• Understand: A simplified fraction represents the same value in a clearer, more concise way.
• Compare: Comparing simplified fractions is much simpler than comparing complex ones.
• Calculate: Calculations involving simplified fractions are generally faster and less prone to errors.

Practice Problems

Try simplifying these fractions using the techniques described above:

• 12/18
• 24/36
• 35/49

By understanding the concept of the greatest common factor and applying the steps outlined, you can confidently simplify any fraction. Remember, practice makes perfect!