Problem 27 - Beginning and Intermediate Algebra

Question

Multiply. $$ \frac{2}{3}\left(-\frac{4}{9}\right) $$

Step-by-Step Solution

Answer

The product is \( \frac{-8}{27} \).

Step 1: Understand the Multiplication of Fractions

When multiplying fractions, multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator. In this case, we have \( \frac{2}{3} \) and \( -\frac{4}{9} \).

Step 2: Multiply the Numerators

Multiply the numerators: \( 2 \times (-4) = -8 \). So, the new numerator is \(-8\).

Step 3: Multiply the Denominators

Multiply the denominators: \( 3 \times 9 = 27 \). So, the new denominator is \(27\).

Step 4: Form the Resulting Fraction

Combine the new numerator and denominator to form the fraction: \( \frac{-8}{27} \).

Step 5: Simplify the Fraction

Check if the fraction can be simplified. In this case, \( \frac{-8}{27} \) is already in its simplest form because \(8\) and \(27\) have no common factors other than 1.

Key Concepts

Numerator

When working with fractions, the numerator is the top number. It represents the number of parts being considered. In the fraction \( \frac{2}{3} \), the numerator is \( 2 \). Similarly, in \( -\frac{4}{9} \), the numerator is \(-4\). The concept of the numerator is crucial because it helps determine how many parts out of the whole are being dealt with.

When multiplying fractions, focus first on the numerators:

• Multiply the numerators of both fractions together.
• This results in the numerator of the new fraction.

In our example, we multiply \( 2 \) by \( -4 \), resulting in \(-8\). This new number forms part of the solution: \( \frac{-8}{27} \). The numerator indicates that there are \(-8\) parts of something that is divided into \(27\) parts.

Denominator

The denominator is the bottom part of a fraction. It shows how many parts the whole is divided into. Taking the fraction \( \frac{2}{3} \) as an example, the denominator is \(3\). For \( -\frac{4}{9} \), it is \(9\).

Understanding the denominator is key, because:

• It indicates the total number of equal parts making up a whole.
• It affects how each part is sized relative to the whole.

When multiplying fractions, multiply the denominators to form the denominator of the new fraction. For our given problem, we multiply \(3\) and \(9\) to get \(27\). This number serves as the denominator in the result \( \frac{-8}{27} \). The denominator remains consistent during operations to reflect its role in dividing the whole into equal sections.

Simplest Form

The simplest form of a fraction is when its numerator and denominator have no common factors other than 1. It represents the fraction in its most reduced form. It's important because it makes fractions easier to understand and simplifies the arithmetic process.

To simplify a fraction:

• Identify any common factors between the numerator and the denominator.
• Divide both the numerator and denominator by their greatest common factor (GCF).
• Repeat the process until no further simplification is possible.

In the exercise, the fraction \( \frac{-8}{27} \) was examined to check for any simplification. Since the numbers \(8\) and \(27\) do not share any common factors except for 1, it is already in its simplest form. This step helps ensure that an improper representation doesn't make the computation more cumbersome than necessary.