Problem 78 - Prealgebra

Question

Write each fraction as an equivalent fraction with denominator 24. $$\frac{1}{8}$$

Step-by-Step Solution

Answer

\( \frac{1}{8} = \frac{3}{24} \)

Step 1: Understand the Proposed Change

We need to convert \( \frac{1}{8} \) to a fraction with a denominator of 24.

Step 2: Find the Multiplication Factor

Determine what number you must multiply the denominator 8 by to get 24. This is done by dividing 24 by 8: \( \frac{24}{8} = 3 \). Thus, you must multiply both the numerator and the denominator by 3.

Step 3: Apply the Multiplication to Both Numerator and Denominator

Multiply both the numerator (1) and the denominator (8) of the original fraction by the factor found in Step 2. \( 1 \times 3 = 3 \) and \( 8 \times 3 = 24 \).

Step 4: Write the Equivalent Fraction

After performing the multiplication, the new fraction is \( \frac{3}{24} \). This is equivalent to \( \frac{1}{8} \) but with the desired denominator of 24.

Key Concepts

Fraction Conversion

Converting a fraction to another with a different denominator is a vital skill in math. It involves changing the appearance of a fraction without changing its value. This process is known as fraction conversion. Let's take a simple fraction, like \( \frac{1}{8} \), and convert it to a different form but with a specific denominator, say 24. This doesn't change the quantity that the fraction represents; it merely changes the way it's expressed. To perform fraction conversion, follow these simple steps:

• Identify the desired denominator. In our case, it is 24.
• Determine the multiplication factor by dividing the desired denominator by the original denominator.
• Multiply both the numerator and the original denominator by this factor.

This makes sure the value of the fraction remains the same even as its form changes.

Finding Common Denominator

Finding a common denominator is crucial when working with multiple fractions, as it allows addition, subtraction, or comparison.For example, converting \( \frac{1}{8} \) to a fraction with a desired denominator of 24, involves a simple process:

• Divide the desired denominator (24) by the original denominator (8) to find the so-called multiplication factor. In our case, \( \frac{24}{8} = 3 \).
• Multiply both the numerator and denominator of the original fraction by this factor, resulting in an equivalent fraction with the new denominator.

This technique is essential for ensuring that fractions are compatible for further arithmetic operations. It harmonizes different fractions into a single common denominator, allowing for simple calculations.

Numerator and Denominator

Fractions are parts of a whole and are made up of two key components: the numerator and the denominator.The numerator is the top number of a fraction that tells you how many parts you have. For example, in \( \frac{1}{8} \), 1 is the numerator indicating one part.The denominator is the bottom number that indicates how many equal parts the whole is divided into. In \( \frac{1}{8} \), the denominator 8 shows the whole is split into eight equal parts. When converting fractions like \( \frac{1}{8} \) to an equivalent form with a different denominator, both components need to be adjusted properly. Finding a multiplication factor helps adjust both the numerator and the denominator. Here's what you do:

• Multiply the numerator by the same factor used to change the denominator, keeping the quantity equal.
• Apply the multiplication uniformly to maintain the balance of the fraction.

Understanding these elements ensures you correctly transform fractions without altering their value.