33 72 simplified

Charlotte

2 min read

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01-03-2025

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Understanding how to simplify fractions is a fundamental skill in mathematics. This guide will walk you through simplifying the fraction 33/72, explaining the process step-by-step and offering insights into simplifying fractions in general. We'll cover finding the greatest common divisor (GCD) and alternative methods. By the end, you'll be able to confidently simplify similar fractions.

Understanding Fraction Simplification

Simplifying a fraction, also known as reducing a fraction to its lowest terms, means finding an equivalent fraction where the numerator and denominator have no common factors other than 1. This makes the fraction easier to understand and work with. The core principle involves dividing both the numerator and denominator by their greatest common divisor (GCD).

Finding the GCD of 33 and 72

To simplify 33/72, we first need to find the greatest common divisor (GCD) of 33 and 72. There are several ways to do this:

Method 1: Listing Factors

1. List the factors of 33: 1, 3, 11, 33
2. List the factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
3. Identify the greatest common factor: The largest number that appears in both lists is 3. Therefore, the GCD of 33 and 72 is 3.

Method 2: Prime Factorization

1. Find the prime factorization of 33: 3 x 11
2. Find the prime factorization of 72: 2 x 2 x 2 x 3 x 3 (or 2³ x 3²)
3. Identify common prime factors: Both numbers share one factor of 3.
4. Calculate the GCD: The GCD is the product of the common prime factors, which is 3.

Method 3: Euclidean Algorithm (for larger numbers)

The Euclidean algorithm is a more efficient method for finding the GCD of larger numbers. It involves repeatedly applying the division algorithm until the remainder is 0. The last non-zero remainder is the GCD.

1. Divide 72 by 33: 72 = 2 x 33 + 6
2. Divide 33 by the remainder 6: 33 = 5 x 6 + 3
3. Divide 6 by the remainder 3: 6 = 2 x 3 + 0

The last non-zero remainder is 3, so the GCD of 33 and 72 is 3.

Simplifying 33/72

Now that we know the GCD is 3, we can simplify the fraction:

33/72 = (33 ÷ 3) / (72 ÷ 3) = 11/24

Therefore, the simplified form of 33/72 is 11/24.

Checking Your Work

Always check your simplified fraction to ensure it's in its lowest terms. 11 and 24 have no common factors other than 1, confirming that 11/24 is the simplest form.

Simplifying Fractions: A General Approach

The process for simplifying any fraction is the same:

1. Find the GCD of the numerator and denominator using any of the methods above.
2. Divide both the numerator and the denominator by the GCD.
3. Check to ensure the resulting fraction is in its lowest terms.

This comprehensive guide demonstrates how to effectively simplify the fraction 33/72 and provides a general framework for simplifying other fractions. Remember to choose the method for finding the GCD that best suits the numbers you're working with. Practice will make you proficient in simplifying fractions quickly and accurately.