91Ó°ÊÓ

To simplify a fraction, the first step is to determine the greatest common divisor (GCD) of both the numerator and the denominator. The GCD is the largest number that can exactly divide both numbers without leaving any remainder. For instance, in our exercise where the fraction is \(-\frac{108}{63}\), we need to find the GCD of 108 and 63.

The GCD of 108 and 63 is calculated by identifying the common factors of both numbers. For example, the common factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 36, 54, 108, and for 63, the factors are 1, 3, 7, 9, 21, 63. The highest common factor from these sets is 9. That's why 9 is the GCD.

To really help in fraction simplification, it's crucial to understand what the numerator and denominator are. Simply put, in a fraction like \(-\frac{108}{63}\), the numerator is the number above the line (108), while the denominator is the number below the line (63).

In any fraction, the numerator represents how many parts we have, and the denominator represents how many equal parts the whole is divided into. When simplifying fractions, we are essentially looking for a way to express the same relationship using smaller, simpler terms. In this example, once we determine the GCD (which is 9), both the numerator and the denominator are divided by this number to simplify the fraction.

Fraction reduction or simplifying fractions involves converting a fraction to its simplest form. This is done by dividing both the numerator and the denominator by their GCD. Using our exercise example \(-\frac{108}{63}\):

• First, identify the GCD of 108 and 63, which is 9.
• Next, divide both the numerator and the denominator by their GCD.

Performing this step:
\(-\frac{108}{63} = -\frac{108 \, \text{div} \, 9}{63 \, \text{div} \, 9} = -\frac{12}{7}\)
Therefore, the simplified form of the fraction \(-\frac{108}{63}\) is \(-\frac{12}{7}\).

Simplifying fractions makes it easier to work with them in mathematical operations and helps in getting an accurate interpretation of the relationship between the values.