91Ó°ÊÓ

Understanding the roles of numerators and denominators is essential in the world of fractions. A fraction represents a part of a whole, and it's composed of two parts: the numerator and the denominator.

The numerator, which is the top number in a fraction, indicates how many parts we are concerned with, while the denominator, the bottom number, tells us how many of those parts make up a whole. For instance, in the fraction \(\frac{3}{7}\), the numerator is 3, meaning we have three parts, and the denominator is 7, indicating that there are seven equal parts in total.

When multiplying fractions, as in the given exercise, we multiply the numerators together to find the new numerator, and we do the same for the denominators to find the new denominator. This process does not involve cross-multiplication or any other operations beyond straightforward multiplication.

Working with negative numbers in fractions follows the same rules as with integers. A negative sign can be associated with the numerator, the denominator, or the entire fraction.

When a fraction has a negative sign, either in the numerator or the denominator, it indicates that the fractional value is less than zero. This also implies that when we multiply a positive number by a negative one, the product is negative. For example, in our exercise \(3 \times -14 = -42\), since 3 is positive and -14 is negative, the resulting numerator is -42.

It's crucial to remember that a negative sign before the fraction or in the numerator is equivalent to having it in the denominator. In other words, \( -\frac{a}{b} \), \( \frac{-a}{b} \), and \( \frac{a}{-b} \) all represent the same value.

Simplifying a fraction, also known as reducing it to its lowest terms, involves dividing the numerator and the denominator by their greatest common divisor (GCD). The purpose of simplification is to make the fraction as simple as possible without changing its value.

To simplify \( -\frac{42}{105} \), we find the GCD of 42 and 105, which is 21. Dividing both the numerator and the denominator by 21, we get \( -\frac{2}{5} \). This simplified fraction is equivalent to the original fraction but is easier to comprehend and work with in further mathematical operations.