Problem 15 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{5}{8} \cdot \frac{4}{15} $$

Step-by-Step Solution

Answer

\(\frac{1}{6}\)

Step 1: Multiply the Numerators

To multiply the fractions, start by multiplying the numerators. Multiply the numerators together: \(5 \times 4 = 20\).

Step 2: Multiply the Denominators

Next, multiply the denominators of the fractions together. Multiply the denominators: \(8 \times 15 = 120\).

Step 3: Form the New Fraction

Form a new fraction using the results from the previous steps: the numerator is \(20\) and the denominator is \(120\). Thus, the fraction is \(\frac{20}{120}\).

Step 4: Simplify the Fraction

To simplify the fraction, find the greatest common divisor (GCD) of \(20\) and \(120\). The GCD is \(20\). Divide both the numerator and the denominator by \(20\). So, \(\frac{20}{120} = \frac{1}{6}\).

Key Concepts

Simplifying Fractions

One of the most important skills when working with fractions is knowing how to simplify them. Simplifying a fraction means expressing it in its simplest form, where the numerator (top number) and the denominator (bottom number) have no common factors other than 1. Here’s how you can do it:

• Identify any common factors shared by the numerator and the denominator.
• Divide both the numerator and the denominator of the fraction by their Greatest Common Divisor (GCD). This reduces the fraction without changing its value.

Simplifying makes it easier to work with fractions, especially when performing further calculations. In the exercise where the product was \( \frac{20}{120} \), finding the GCD (20) allowed us to simplify the fraction to \( \frac{1}{6} \). This smaller fraction is equivalent to the original and is much simpler to interpret.

Numerators and Denominators

Understanding numerators and denominators is crucial for working with fractions.

• The **numerator** is the top number of a fraction. It tells how many parts of the whole are being considered.
• The **denominator** is the bottom number. It indicates the total number of equal parts that make up the whole.

This conceptual framework helps in performing operations such as multiplication, where we multiply numerators together and denominators together, as seen in the step by step solution of \( 5 \times 4 = 20 \) for the numerators and \( 8 \times 15 = 120 \) for the denominators. By understanding these components, you can easily perform arithmetic involving fractions. You will be more adept at visualizing what each fraction represents, which is particularly useful for further operations like addition or subtraction.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is a valuable mathematical tool for simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator without leaving a remainder. By using the GCD, you simplify fractions to their smallest form.How to find the GCD:

• List the factors of both the numerator and the denominator.
• Identify the largest factor that occurs in both lists.
• Ensure to divide both the numerator and the denominator by this factor to simplify the fraction.

In our exercise, finding the GCD of \(20\) and \(120\) (which is \(20\)) allowed us to transform \(\frac{20}{120}\) into \(\frac{1}{6}\). Once you master finding the GCD, simplifying fractions becomes much easier and more intuitive, helping to uncover the simplest form of any fraction quickly.