Problem 69 - Prealgebra

Question

Multiply. $$\left(-\frac{5}{4}\right)\left(-\frac{4}{5}\right)$$

Step-by-Step Solution

Answer

The result of multiplying is 1.

Step 1: Multiply the Numerators

To multiply two fractions, first multiply the numerators. The numerators in this case are -5 and -4. When we multiply these, we get: \(-5 \times -4 = 20\).

Step 2: Multiply the Denominators

Next, multiply the denominators of the fractions. The denominators here are 4 and 5. So, we have: \(4 \times 5 = 20\).

Step 3: Form the New Fraction

Combine the results from Steps 1 and 2 to form the new fraction. This gives us \(\frac{20}{20}\).

Step 4: Simplify the Fraction

The fraction \(\frac{20}{20}\) equals 1 because both the numerator and the denominator are the same number. So, the fraction simplifies to 1.

Key Concepts

Numerators and Denominators in Fractions

Fractions are a fundamental concept in mathematics, representing parts of a whole. Each fraction has two main components: the numerator and the denominator.
The numerator is the top number in a fraction and represents how many parts we have. Meanwhile, the denominator is the bottom number that tells us into how many parts the whole is divided.
Understanding these parts is crucial for any operation involving fractions, such as addition, subtraction, multiplication, and division. For example, in the fraction \(\frac{3}{5}\), 3 is the numerator and 5 is the denominator. Here, they convey the message that you are dealing with 3 out of 5 parts of a whole.
Recognizing the roles of numerators and denominators helps clarify the magnitude and direction of the fraction, especially when dealing with operations like multiplication.

Simplifying Fractions

Simplifying a fraction means reducing it to its simplest form. This process makes it easier to understand and often quicker to use in calculations.
When a fraction is simplified, the numerator and denominator share no common factors other than 1. To simplify a fraction, divide both the numerator and the denominator by their greatest common factor (GCF).
For instance, the fraction \(\frac{8}{12}\) can be simplified. First, find that the GCF of 8 and 12 is 4.
Now divide both 8 and 12 by 4, simplifying the fraction to \(\frac{2}{3}\).
This simplification step also applies after multiplying fractions. Just like the example \(\frac{20}{20}\), where both the numerator and denominator are the same, resulting in a simplified fraction of 1. It is vital to simplify your final answer for clarity and accuracy.

Multiplication of Negative Numbers

Multiplying negative numbers follows a straightforward rule: the product of two negative numbers is positive.
When two numbers have the same sign, whether positive or negative, their product is positive.
This is because the rule for sign multiplication states that a negative times a negative equals a positive.
In the context of fractions, like with \(-\frac{5}{4}\) and \(-\frac{4}{5}\), multiply the numerators and the denominators separately. You'll find both results are positive due to the multiplication rule of negatives:

• Numerators: \(-5 \times -4 = 20\)
• Denominators: \(4 \times 5 = 20\)

Hence, the fraction \(\frac{20}{20}\) simplifies to 1, clearly illustrating how the principles of negative number multiplication apply to fractions.