Problem 41 - Prealgebra

Question

Write each number as an equivalent fraction with denominator \(24 a\). $$2$$

Step-by-Step Solution

Answer

\(\frac{2}{1} = \frac{48a}{24a}\)

Step 1: Understanding the Problem

We need to express the number 2 as a fraction with a denominator of \(24a\). Currently, 2 can be thought of as \(\frac{2}{1}\).

Step 2: Finding the Equivalent Fraction

To convert \(\frac{2}{1}\) into a fraction with denominator \(24a\), we need to multiply both the numerator and the denominator by \(24a\).

Step 3: Performing the Multiplication

Multiply the numerator 2 by \(24a\) to get \(2 \times 24a = 48a\). Similarly, multiply the denominator 1 by \(24a\) to get \(1 \times 24a = 24a\).

Step 4: Writing the Equivalent Fraction

The fraction \(\frac{2}{1}\) is equivalent to \(\frac{48a}{24a}\). This fraction has the desired denominator of \(24a\).

Key Concepts

Understanding the Denominator in Fractions

When learning about fractions, the term "denominator" frequently appears. The denominator is the number below the line in a fraction that tells us into how many equal parts the whole is divided. For example, in the fraction \( \frac{3}{4} \), the denominator is 4, indicating the whole is divided into 4 equal parts.
For a fraction to be valid, the denominator cannot be zero, as division by zero is undefined.

• The denominator informs us about the scale or size of each part. A larger denominator means smaller parts.
• In operations involving fractions, especially when finding common denominators, the denominator plays a key role.

In the original exercise, converting the number 2 into a fraction with a denominator of \(24a\) expands our understanding of the fraction's size and allows comparisons or operations with other fractions sharing the same denominator.

Conversion of Fractions Made Easy

The conversion of fractions involves changing a given fraction to an equivalent fraction with a desired denominator. An equivalent fraction has the same overall value but different numerators and denominators. This process is essential in comparing fractions, adding, or subtracting them.
To convert a fraction to an equivalent one with a specific denominator, both the numerator and the denominator of the original fraction are multiplied by the same number. This operation does not change the fraction's value because multiplying by 1 in a different form (e.g., \( \frac{24a}{24a} \)) retains the fraction's value.

• Start by deciding the desired denominator.
• Multiply both the numerator and denominator by the factor that converts the original denominator into the desired denominator.

For example, to turn \( \frac{2}{1} \) into a fraction with a denominator of \(24a\), multiply by \( \frac{24a}{24a} \). This results in \( \frac{48a}{24a} \), maintaining the same value of 2.

Simplifying Multiplication in Fractions

Multiplication in fractions may sound complicated, but it's a straightforward process. When multiplying a fraction by another number to alter its form, the operation affects both the numerator and the denominator equally. The relationship between these components remains intact, keeping the fraction's value unchanged.
In the exercise, to convert \( \frac{2}{1} \) to have a denominator of \(24a\), the multiplication step is crucial.

• Multiply the numerator by the target conversion factor.
• Do the same with the denominator.

This operation is akin to multiplying by 1, since \( \frac{24a}{24a} = 1 \). Therefore, \( 2 \times \frac{24a}{24a} = \frac{48a}{24a} \), achieving an equivalent fraction with the specified denominator. Understanding this process is essential for fraction operations in mathematics.