72/99 simplified

2 min read 11-03-2025

Simplifying fractions, also known as reducing fractions to their lowest terms, is a fundamental skill in mathematics. It involves finding an equivalent fraction where the numerator and denominator have no common factors other than 1. This guide will walk you through simplifying the fraction 72/99.

Understanding Fraction Simplification

Before we dive into simplifying 72/99, let's review the basics. A fraction represents a part of a whole. It's composed of a numerator (the top number) and a denominator (the bottom number). Simplifying a fraction means dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.

Finding the Greatest Common Divisor (GCD) of 72 and 99

To simplify 72/99, we first need to find the GCD of 72 and 99. There are several ways to do this:

Method 1: Listing Factors

List all the factors of 72 and 99:

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Factors of 99: 1, 3, 9, 11, 33, 99

The largest number that appears in both lists is 9. Therefore, the GCD of 72 and 99 is 9.

Method 2: Prime Factorization

This method involves breaking down each number into its prime factors:

Prime factorization of 72: 2 x 2 x 2 x 3 x 3 = 2³ x 3²
Prime factorization of 99: 3 x 3 x 11 = 3² x 11

The common prime factors are 3² (or 9). Therefore, the GCD is 9.

Method 3: Euclidean Algorithm

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

Divide 99 by 72: 99 = 1 x 72 + 27
Divide 72 by 27: 72 = 2 x 27 + 18
Divide 27 by 18: 27 = 1 x 18 + 9
Divide 18 by 9: 18 = 2 x 9 + 0

The last non-zero remainder is 9, so the GCD is 9.

Simplifying 72/99

Now that we know the GCD is 9, we can simplify the fraction:

72/99 = (72 ÷ 9) / (99 ÷ 9) = 8/11

Therefore, the simplified form of 72/99 is 8/11.

Checking Your Work

It's always a good idea to check your work. You can verify that 8/11 is the simplified form by ensuring that the numerator (8) and the denominator (11) share no common factors other than 1. Since 8 and 11 are relatively prime (they have no common divisors other than 1), our simplification is correct.

Conclusion

Simplifying fractions is a crucial skill in mathematics. By understanding the concept of the greatest common divisor and employing methods like prime factorization or the Euclidean algorithm, you can efficiently reduce fractions to their lowest terms. As demonstrated, 72/99 simplifies to the much more manageable fraction 8/11. Remember to always check your answer to ensure accuracy.