Problem 13 - Algebra A Combined Function

Question

Write each fraction as an equivalent fraction with the given denominator. See Example 7 . $$ \frac{2}{9} \text { with a denominator of } 18 $$

Step-by-Step Solution

Answer

\( \frac{2}{9} \) is equivalent to \( \frac{4}{18} \).

Step 1: Identify the Original Fraction

The original fraction is \( \frac{2}{9} \).

Step 2: Determine the Required Denominator

The fraction needs to be expressed with a denominator of 18.

Step 3: Find the Conversion Factor

To convert \( \frac{2}{9} \) to a fraction with denominator 18, determine the conversion factor by dividing the new denominator by the original denominator. Calculate \( \frac{18}{9} = 2 \).

Step 4: Apply the Conversion Factor to the Numerator

Multiply the numerator of the original fraction by the conversion factor: \( 2 \times 2 = 4 \).

Step 5: Write the Equivalent Fraction

The equivalent fraction with a denominator of 18 is \( \frac{4}{18} \).

Key Concepts

Understanding Denominators

In fractions, the denominator is the bottom number, indicating how many equal parts the whole is divided into. Understanding denominators is crucial because they determine the size of each part relative to the whole.
For example, in the fraction \( \frac{2}{9} \), the denominator is 9, meaning the whole is divided into 9 equal parts, and you have two of these parts.
Denominators must always be non-zero, as dividing by zero is undefined in mathematics.

• A larger denominator means smaller individual parts, as you are dividing the whole into more pieces.
• A smaller denominator means larger individual parts.
• Matching denominators are required for operations like addition and subtraction of fractions.

Knowing how to work with denominators allows for converting and comparing fractions easily.

The Role of the Conversion Factor

The conversion factor is the key to changing a fraction's denominator without altering its value. It is found by dividing the desired denominator by the original denominator.
In our example, to get from a denominator of 9 to 18, the conversion factor is \( \frac{18}{9} = 2 \). This tells us how much each part of the fraction needs to be multiplied to stay equivalent.
Using the conversion factor involves:

• Finding the conversion factor by division, ensuring it's a whole number for the fraction to stay valid.
• Multiplying both the numerator and denominator by the conversion factor. This keeps the fraction equivalent.

The conversion factor is crucial in creating equivalent fractions, transforming fractions to have the same denominators necessary for comparison or arithmetic operations.

What Are Equivalent Fractions?

Equivalent fractions represent the same value or proportion of the whole, even if their numerators and denominators differ. Fractions like \( \frac{2}{9} \) and \( \frac{4}{18} \) are equivalent because they describe the same portion of a whole.
Establishing equivalent fractions depends on the conversion factor, which ensures that as we change the denominator, the overall fraction value stays the same.

• Equivalent fractions have different names but represent the same point on a number line.
• They are essential for operations that require common denominators, such as adding or comparing fractions.
• To find them, multiply or divide both numerator and denominator by the same non-zero number.

Equivalent fractions play a significant role in simplifying fractions and solving complex fraction problems efficiently.