Problem 76 - Prealgebra

Question

Write each fraction as an equivalent fraction with denominator 24. $$\frac{1}{4}$$

Step-by-Step Solution

Answer

The fraction \( \frac{1}{4} \) as an equivalent fraction with denominator 24 is \( \frac{6}{24} \).

Step 1: Find Common Denominator

To find an equivalent fraction with a specific denominator, we must convert the given fraction so that it has the desired denominator. In this case, the desired denominator is 24.

Step 2: Determine Multiplier

To convert the fraction \( \frac{1}{4} \) to a fraction with a denominator of 24, find the number that 4 must be multiplied by to become 24. This is done by dividing 24 by 4: \( 24 \div 4 = 6 \).

Step 3: Modify the Fraction

Multiply both the numerator and the denominator of \( \frac{1}{4} \) by 6 to find the equivalent fraction: \( \frac{1 \times 6}{4 \times 6} = \frac{6}{24} \).

Step 4: Verify the Fraction

Check that the new fraction \( \frac{6}{24} \) has the same denominator as required, which it does. Also, verify that the calculation follows the rules of equivalent fractions properly.

Key Concepts

Understanding Fractions

Fractions are a way to represent parts of a whole. They consist of two numbers separated by a line. The number on the top is called the *numerator*, and the number on the bottom is the *denominator*.
The numerator represents how many parts we are considering, while the denominator indicates how many parts the whole is divided into. For example, in the fraction \( \frac{1}{4} \), the number 1 is the numerator, and it signifies a single part out of four equal parts that make up a whole.

• *Numerator* - Tells us the number of parts taken.
• *Denominator* - Tells us the total number of equal parts the whole is divided into.

Fractions are not only limited to representing parts smaller than a whole but can also be used for numbers greater than or equal to one; these are usually called improper fractions or mixed numbers when combined with whole numbers.

Finding a Common Denominator

One of the first steps in creating equivalent fractions is to ensure they have a common denominator. A common denominator is a shared multiple or common base, allowing fractions to be compared or combined easily.
For the fraction \( \frac{1}{4} \), changing the denominator to 24 required us to determine a number we can multiply the original denominator (which is 4) by to reach 24. This step is critical in many fraction operations like addition or comparison.

• *Common Denominator* - Makes it possible to add, subtract, or compare fractions with different denominators.
• To find it, use multiplication, division, or both, depending on what is needed.

This concept of common denominators is essential in achieving a harmony between fractions, turning operations like addition and subtraction into smooth processes.

The Multiplication of Fractions

Multiplying fractions is a straightforward process. It involves multiplying the numerators with each other and the denominators with each other. However, when adjusting fractions to have a specific denominator, multiplication serves as a tool for scaling or finding equivalents.
In our example, to transform \( \frac{1}{4} \) into \( \frac{6}{24} \), we determined that multiplying by 6 would adjust both the numerator and the denominator appropriately. This multiplication yields the equivalent fraction where the value does not change, only its appearance.

• Multiply the numerators across for a new numerator.
• Multiply the denominators across for a new denominator.

This technique is often used alongside division for simplifying or scaling up fractions, allowing them to fit required contexts seamlessly.