Problem 44 - Prealgebra

Question

Write each number as an equivalent fraction with denominator \(24 a\). $$8$$

Step-by-Step Solution

Answer

The number 8, expressed as a fraction with denominator \(24a\), is \(\frac{192a}{24a}\).

Step 1: Express the Number as a Fraction

Any whole number can be written as a fraction with a denominator of 1. Thus, you can express 8 as \(\frac{8}{1}\).

Step 2: Find the New Denominator

We need to write the fraction with a new denominator of \(24a\). Currently, our fraction \(\frac{8}{1}\) has a denominator of 1. Our goal is to change it to \(24a\).

Step 3: Determine the Multiplier for the Denominator

To convert the denominator from 1 to \(24a\), we must multiply by \(24a\). So the multiplier is \(24a\).

Step 4: Multiply Both the Numerator and Denominator by the Multiplier

To maintain the fraction's value, multiply both the numerator and the denominator by \(24a\). This gives us:\[\frac{8 \times 24a}{1 \times 24a} = \frac{192a}{24a}\]

Step 5: Simplify the Fraction if Possible

Although typically you would check to simplify the fraction, since the denominator requested is \(24a\), no further simplification is required. The fraction remains \(\frac{192a}{24a}\).

Key Concepts

Fractions

A fraction is a way of expressing a part of a whole or a division of quantities in mathematics. It consists of two parts: a numerator and a denominator.
The numerator is the top number, and it represents how many parts we have or are considering. The denominator is the bottom number, indicating the total number of equal parts the whole is divided into.
Fractions can be used to show amounts less than one (proper fractions), equal to one (unity), or even more than one (improper fractions). For example, we expressed the number 8 as the fraction \(\frac{8}{1}\), meaning 8 whole parts.

• Proper Fractions: Numerator is less than the denominator.
• Improper Fractions: Numerator is equal to or greater than the denominator.
• Equivalent Fractions: Different fractions that represent the same value.

Understanding fractions is crucial because they are used widely in various fields, including science, engineering, and everyday life.

Denominator

The denominator is a key part of a fraction as it indicates into how many parts the whole is divided.
In our exercise, we started by expressing the whole number 8 as \(\frac{8}{1}\). This meant 8 whole units, with the denominator being 1.
To find an equivalent fraction with a different denominator, we change the bottom number while making sure the fraction still represents the same quantity.
Here we aimed to transform \(\frac{8}{1}\) into a fraction with a denominator of \(24a\). This means we need a common scale, which helps us understand the concept of equal parts represented differently. Maintaining the fraction's value requires adjusting the numerator alongside the denominator.

Numerator Transformation

Numerator transformation is essential when converting fractions to an equivalent form with a new denominator.
To maintain the value of a fraction, both the numerator and the denominator must be multiplied by the same amount.
In the given exercise, we transformed \(\frac{8}{1}\) to have a denominator of \(24a\) by multiplying both the numerator and the denominator by \(24a\).

• Original fraction: \(\frac{8}{1}\)
• Multiplier: \(24a\)
• Transformed fraction: \(\frac{8 \times 24a}{1 \times 24a} = \frac{192a}{24a}\)

Such transformations ensure fractions retain their original value, making them useful for comparisons or calculations with different denominators.