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Dealing with negative fractions might seem tricky at first, but once you understand the basics, it becomes quite straightforward. A negative fraction simply means that the fraction as a whole has a negative value. This could be represented with the negative sign in front of the numerator, like \( -\frac{3}{4} \) or in front of the denominator like \( \frac{3}{-4}\), or even before the entire fraction like \( -(\frac{3}{4})\). Importantly, no matter where the negative sign is placed, it affects the fraction equally.

When multiplying two negative fractions, such as \( (-\frac{3}{8})(-\frac{4}{15})\), the negatives cancel each other out. This is because the rule of multiplication states that a negative times a negative yields a positive. Hence, \( -\frac{3}{8} \) times \( -\frac{4}{15} \) gives us a positive fraction \( \frac{3}{8} \) times \( \frac{4}{15}\), which we then multiply the numerators and denominators separately to find the product.

Simplifying fractions is a critical step to ensure that the answer is in its most reduced form. The goal is to divide both the numerator and the denominator by a common number until no common divisor exists other than 1.

As shown in the exercise, the initial result of multiplying \( -\frac{3}{8} \) and \( -\frac{4}{15} \) is \( \frac{12}{120}\), but this fraction is not in its simplest form. To simplify, we need to find the greatest common divisor (GCD) - the largest number that divides evenly into both numerator and denominator.

Simplification Process

In our example, 12 divides both the numerator and the denominator, thus, dividing both by 12 gives us the simplified fraction \( \frac{1}{10}\). Always simplify to improve clarity and understanding of the fraction's actual value.

When you tackle problems involving fractions, understanding the concept of the greatest common divisor (GCD) is incredibly useful. The GCD of two numbers is the largest number that can divide both numbers without leaving a remainder. Identifying and using the GCD is instrumental in simplifying fractions to their most basic form, which is usually required to complete many mathematical exercises.

For instance, upon multiplying the negative fractions \( -\frac{3}{8} \) and \( -\frac{4}{15}\) and getting the result \( \frac{12}{120}\), the next step is to simplify this fraction. To do so, one needs to find the GCD of 12 and 120, which is 12, and then divide both numerator and denominator by this number. This process results in the final, simplified fraction of \( \frac{1}{10}\). Knowing how to determine the GCD not only helps with simplifying fractions but is also useful in other aspects of mathematics, such as solving equations and finding least common multiples.