Problem 109 - Introductory and Intermediate Algebra for College Students 4th

Question

Will help you prepare for the material covered in the next section. In each exercise, use the given formula to perform the indicated operation with the two fractions. $$\frac{a}{b} \cdot \frac{c}{d}=\frac{a \cdot c}{b \cdot d} ; \quad \frac{3}{7} \cdot \frac{2}{5}$$

Step-by-Step Solution

Answer

The result of \( \frac{3}{7} \cdot \frac{2}{5} \) is \( \frac{6}{35} \).

Step 1: Understanding the given formula

The given formula for multiplication of fractions is: \( \frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \). This formula means that to multiply two fractions, we multiply the numerators (the top parts) to get the new numerator, and we multiply the denominators (the bottom parts) to get the new denominator.

Step 2: Apply the formula

Given \(\frac{3}{7}\) and \(\frac{2}{5}\). Apply the formula: The new numerator is \(3 \cdot 2 = 6\), and the new denominator is \(7 \cdot 5 = 35\). Thus, \(\frac{3}{7} \cdot \frac{2}{5} = \frac{6}{35}\).

Step 3: Simplify the result

The fraction \(\frac{6}{35}\) is already in its simplest form, as 6 and 35 do not share any common factors other than 1. Therefore, the fraction remains as \(\frac{6}{35}\).

Key Concepts

Understanding the Numerator

The numerator is the number found at the top of a fraction. It's a crucial part of fraction multiplication. When multiplying fractions, the numerators from each fraction are multiplied together to form the new numerator in the resulting fraction.
For example, consider the multiplication of \(\frac{3}{7}\) and \(\frac{2}{5}\). Here, the numerators are 3 and 2.

• Multiply them together: \(3 \times 2 = 6\)
• This gives us the new numerator in the product fraction

Understanding the numerator helps you accurately carry out multiplication operations with fractions.

Understanding the Denominator

The denominator is the number located at the bottom of a fraction. It represents the total number of equal parts the whole is divided into.
In fraction multiplication, you multiply the denominators together to get the new denominator.
Using the example \(\frac{3}{7}\) and \(\frac{2}{5}\), the denominators 7 and 5 are multiplied:

• Carry out the multiplication: \(7 \times 5 = 35\)
• This results in 35 as the denominator in the final fraction

The denominator holds the key to determining the division of a whole into smaller parts, thus aiding in visualizing fractions.

Simplifying Fractions

Simplifying a fraction means adjusting it to its simplest form where the numerator and the denominator only share a factor of one. It's important for clarity in both math exercises and practical applications.

• Identify any common factors between the numerator and denominator
• Divide both terms by their greatest common factor

Looking at \(\frac{6}{35}\), we check for any common factors between 6 and 35:

• Since the only common factor is 1, \(\frac{6}{35}\) is already simplified

Simplifying helps make fractions easier to understand and work with by reducing them to their most basic form.