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When we simplify fractions, a key step is finding the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest number that can evenly divide both numbers. To find the GCD of 63 and 84, we need to look for the highest number that divides both without leaving a remainder.
One way to identify the GCD is by listing the factors of each number:

• Factors of 63: 1, 3, 7, 9, 21, 63
• Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

Among these factors, the highest common number is 21. So, 21 is the GCD of 63 and 84.
Understanding the GCD helps us reduce fractions to their simplest form. By dividing both the numerator and the denominator by their GCD, we can effectively simplify the fraction. This concept is essential whenever you're working with fraction reduction.

In a fraction, the top number is the numerator, and the bottom number is the denominator. These two components are critical for understanding fractions and simplifying them.

• The numerator represents how many parts we have.
• The denominator indicates how many parts make up a whole.

For example, in the fraction \(-\frac{63}{84}\), 63 is the numerator and 84 is the denominator.
When simplifying, you'll need to consider both the numerator and the denominator to find their GCD and divide them both accordingly.
This process ensures that the fraction remains equivalent, only in a simpler form—that means it doesn't change the value of the fraction, just how it's expressed.

Fraction reduction is the process of simplifying a fraction to its lowest terms. This is done by dividing both the numerator and the denominator by their GCD.
Let's go through the reduction for \(-\frac{63}{84}\):

• Find the GCD of 63 and 84, which is 21.
• Divide the numerator (63) by the GCD: 63 ÷ 21 = 3.
• Divide the denominator (84) by the GCD: 84 ÷ 21 = 4.

So, \(-\frac{63}{84} = -\frac{3}{4}\).
This tells us that \(-\frac{63}{84}\) simplifies to \(-\frac{3}{4}\) after fraction reduction.
Reducing fractions makes them easier to work with and understand, especially when performing operations like addition, subtraction, multiplication, or division.