Problem 42 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{3}{7} \cdot \frac{1}{4}$$

Step-by-Step Solution

Answer

The result of the operation \(\frac{3}{7} \cdot \frac{1}{4}\) is \(\frac{3}{28}\)

Step 1: Multiplication of Fractions

To multiply two fractions, you should multiply the numerators with each other and the denominators with each other. So, the multiplication of \(\frac{3}{7}\) by \(\frac{1}{4}\) is done by multiplying the numerators 3 and 1, and the denominators 7 and 4. The result is \(\frac{3*1}{7*4}\) which simplifies to \(\frac{3}{28}\)

Step 2: Simplification

The next step is to reduce the fraction to its lowest terms, if it is possible. A fraction is in its lowest terms if the only common factor of the numerator and the denominator is 1. The Euclidean Algorithm or Prime Factorization can be used to check for common factors. Looking at the result of our multiplication in Step 1, \(\frac{3}{28}\), there are no common factors between the numerator 3 and the denominator 28, other than 1. Thus, the fraction \(\frac{3}{28}\) is already in its lowest terms and doesn't require further simplification.

Key Concepts

Lowest Terms

When working with fractions, putting them in their lowest terms makes them simpler and easier to understand. A fraction is in its lowest terms when the greatest common divisor (GCD) of its numerator and denominator is 1. This means they have no common factors other than 1.

To check if a fraction is simplified, you can follow these steps:

• Find the GCD of the numerator and denominator.
• If the GCD is greater than 1, divide both the numerator and the denominator by this number.
• If the GCD is 1, the fraction is already in its simplest form.

For example, in \(\frac{3}{28}\), 3 and 28 have no common factors, so the fraction is already simplified.

Numerator

The numerator is the top part of a fraction and represents the number of parts being considered. When you multiply fractions, just multiply the numerators together.

For example, if you have \(\frac{3}{7}\) and \(\frac{1}{4}\), the numerators are 3 and 1. Multiply them to get:

• \(3 \times 1 = 3\)

This product becomes the numerator of the new fraction. In this exercise, the resulting numerator was 3.

Denominator

The denominator is the bottom part of a fraction and indicates the total number of equal parts. A fundamental rule in fraction multiplication is that you multiply the denominators of both fractions.

Using \(\frac{3}{7}\) and \(\frac{1}{4}\) as examples, the denominators are 7 and 4. You multiply them to form the new denominator:

• \(7 \times 4 = 28\)

So in the final fraction \(\frac{3}{28}\), 28 is the denominator.

Simplification

Simplification is the process of reducing a fraction to its lowest terms. It makes calculations easier and results clearer.

To simplify a fraction:

• Identify the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by this GCD.

In the task example \(\frac{3}{28}\), since the only common factor between 3 and 28 is 1, no further simplification is possible.
Understanding simplification helps to avoid unnecessary steps and ensures you always have the simplest form at the end of your calculations.