Problem 42 - Beginning and Intermediate Algebra

Question

Write each fraction as an equivalent fraction with the given denominator. See Example 6 . \(\frac{2}{3}\) with a denominator of 9

Step-by-Step Solution

Answer

\(\frac{2}{3} = \frac{6}{9}\) when the denominator is changed to 9.

Step 1: Identify the relationship between the denominators

To make the given fraction have the desired denominator, identify how the original denominator (3) must change to become the new denominator (9). Since 9 is a multiple of 3, specifically 3 multiplied by 3 equals 9, this tells us the factor needed to change the denominator.

Step 2: Adjust the numerator

Once you've determined the factor by which the denominator is multiplied (in this case, 3), multiply the numerator by this same factor to ensure the fraction remains equivalent. For \(\frac{2}{3}\), multiply the numerator 2 by the factor of 3, resulting in a new numerator of 6.

Step 3: Write the equivalent fraction

Using the calculated numerator and the new denominator, write the equivalent fraction. The numerator calculated is 6, and the new denominator is 9, making the equivalent fraction \(\frac{6}{9}\).

Key Concepts

Understanding Fractions

Fractions are a way to express numbers that are not whole. Essentially, a fraction represents a part of a whole. A fraction consists of two parts:

• The numerator, which is the top number.
• The denominator, which is the bottom number.

Together, these components create a ratio that shows how many parts of a certain size are present. For example, in the fraction \(\frac{2}{3}\), the numerator is 2, and the denominator is 3.
Fractions can describe parts of sets, parts of shapes, or portions of objects. They are widely used in mathematics to present non-whole quantities. An important aspect of fractions is understanding how to transform them into equivalent fractions, which helps in comparing and computing with different fractions. This practice is essential when dealing with different denominators or when simplifying fractions.

Role of the Denominator

The denominator is a crucial component of a fraction. It indicates the number of equal parts the whole is divided into. In the fraction \(\frac{2}{3}\), the number 3 signifies that the whole is divided into 3 equal parts. How the whole is split directly affects the size of the parts.
When creating equivalent fractions, adjusting the denominator keeps the same proportion between parts. For example, by changing the denominator from 3 to 9, we are dividing the whole into 9 smaller parts instead of 3. To keep the balance of the fraction, the same multiplication or division principle applies to both the numerator and the denominator. This ensures that the value of the fraction stays the same, even though the numbers are changing. Understanding the role of the denominator aids in grasping how fractions can be equivalent, no matter how they appear numerically.

Importance of the Numerator

The numerator in a fraction represents the number of parts being considered or taken from the whole. In our fraction \(\frac{2}{3}\), the numerator 2 shows that we are looking at 2 out of the 3 total parts. When creating an equivalent fraction, it is crucial to change the numerator in relation to the changes made to the denominator to keep the fraction equivalent.
For instance, to transform \(\frac{2}{3}\) into a fraction with a denominator of 9, we multiply both parts by 3 (since 9 divided by 3 equals 3). This gives us a new numerator of 6, leading to the equivalent fraction \(\frac{6}{9}\). The act of adjusting the numerator keeps the fraction's value consistent while still helping us work with different denominators more conveniently. Understanding the numerator's function allows for more effective manipulation and comparison of fractions.