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Understanding how to multiply fractions is crucial to mastering many aspects of mathematics, as well as real-world applications like measuring and cooking. When multiplying fractions, we take the numerators (the top numbers) of the fractions and multiply them together to get the new numerator. Then, we do the same with the denominators (the bottom numbers), multiplying them to get the new denominator.

For example, let's consider the exercise \(\frac{5}{12}(-\frac{8}{15})\). We multiply the numerators 5 and -8 to get -40, and the denominators 12 and 15 to get 180, forming the new fraction \(\frac{-40}{180}\). But that's not the end of the process—next, we simplify the result, a step we'll explore in more depth in the section on simplifying fractions.

A fraction consists of two parts: the numerator and the denominator. The numerator, located above the fraction bar, represents the number of parts we have. The denominator, below the fraction bar, denotes the total number of equal parts in a whole. When multiplying fractions, each part is treated separately but in a correlated manner.

It's important to pay close attention to the signs of the numerators. If either numerator is negative, the resulting fraction will also be negative. This is evident in our example where we multiplied \(5\) (positive) by \( -8\) (negative), resulting in a negative numerator (-40). Conversely, had both numerators been negative, the result would be positive, as a negative times a negative equals a positive.

Once we've multiplied our fractions and have the new numerator and denominator, the next step is to simplify the fraction to its lowest terms. Simplifying a fraction makes it easier to understand and work with. To do this, we find the greatest common divisor (GCD) of the numerator and denominator and divide both by that number.

Returning to our exercise \(\frac{-40}{180}\), the GCD of 40 and 180 is 20. When you divide both the numerator and the denominator by 20, the result is \(\frac{-2}{9}\). The simplified fraction is both more readable and a truer representation of the fraction's value. Remember, the simplification process does not change the value of the fraction—it merely reduces it to its simplest form.