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The greatest common divisor, often abbreviated as GCD, is a key concept when simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator of a fraction. For example, to simplify the fraction \(\frac{84}{108}\), you need to find the GCD of 84 and 108. The GCD in this case is 12, because 12 is the highest number that divides both 84 and 108 without leaving a remainder.

To find the GCD, you can use different methods like the Euclidean algorithm, prime factorization, or listing the factors for both numbers. Once you have the GCD, you can proceed to simplify your fraction. This step is essential, as it lays the foundation for simplifying fractions accurately.

In any fraction, such as \(\frac{84}{108}\), the top number is called the numerator, and the bottom number is the denominator. The numerator represents how many parts you have, while the denominator shows the number of equal parts the whole is divided into.

Understanding the roles of the numerator and the denominator is essential for fraction simplification. By dividing both the numerator and the denominator by their greatest common divisor (GCD), you make the fraction as simple as possible. For instance, in \(\frac{84}{108}\), 84 is the numerator and 108 is the denominator. When divided by their GCD of 12, we get \(\frac{7}{9}\).

This division does not change the value of the fraction, only how it is expressed. So, the original fraction \(\frac{84}{108}\) and the simplified fraction \(\frac{7}{9}\) are equivalent but the latter is much simpler.

Simplifying fractions makes them easier to understand and work with. It involves reducing the fraction to its simplest form by dividing both the numerator and denominator by their greatest common divisor (GCD).

Let's walk through simplifying \(\frac{84}{108}\) step by step:

• Step 1: Find the GCD of 84 and 108, which is 12.
• Step 2: Divide the numerator (84) by the GCD (12) to get 7.
• Step 3: Divide the denominator (108) by the GCD (12) to get 9.
• Step 4: The fraction is now simplified to \(\frac{7}{9}\).

This process ensures that the fraction is in its simplest form, making it easier to compare, add, subtract, multiply, or divide with other fractions. Simplifying fractions is a fundamental skill in math that helps in solving various types of problems more efficiently.