Problem 84 - Prealgebra

Question

Write each fraction as an equivalent fraction with denominator \(15 x\). $$\frac{2}{x}$$

Step-by-Step Solution

Answer

The equivalent fraction is \( \frac{30}{15x} \).

Step 1: Identify the Target Denominator

We need to write the fraction with the denominator 15x. The target denominator is given as \(15x\).

Step 2: Set the Fraction Equal to the Equivalent Form

To find the equivalent fraction, start by setting \( \frac{2}{x} \) equal to \( \frac{N}{15x} \), where \( N \) is the unknown numerator that we need to find.

Step 3: Determine the Multiplying Factor

To transform the denominator from \( x \) to \( 15x \), multiply both the numerator and the denominator of the original fraction by what is needed. The needed factor is \( \frac{15x}{x} = 15 \).

Step 4: Calculate the New Numerator and Fraction

Multiply both the numerator and the denominator of \( \frac{2}{x} \) by 15: \( \frac{2 \times 15}{x \times 15} = \frac{30}{15x} \).

Step 5: Confirm the Fraction Equality

Check that \( \frac{30}{15x} \) is indeed equivalent to \( \frac{2}{x} \) when simplified back: \( 30 \div 15 = 2 \) and \( 15x \div 15 = x \). Thus, they are equivalent.

Key Concepts

Understanding Fractions

Fractions are a way to represent a part of a whole. They are written in the form \( \frac{a}{b} \), where \( a \) is the numerator and \( b \) is the denominator.
- The numerator (top number) shows how many parts are taken from the whole.
- The denominator (bottom number) shows how many equal parts the whole is divided into.
A fraction can be understood as a division of the numerator by the denominator.
For example, \( \frac{2}{x} \) tells us that we have 2 parts out of \( x \) total parts. In fractions, the larger the denominator, the smaller each part becomes.
Equivalent fractions are different fractions that represent the same value.
To determine if two fractions are equivalent, you can simplify or find a common factor that changes one fraction into the other.
These fractions depend on multiplying or dividing the numerator and denominator by the same number.

The Role of the Denominator

The denominator in a fraction reveals how many equal pieces the whole item is divided into. For the fraction \( \frac{2}{x} \), \( x \) represents the number of equal parts.
When converting fractions to equivalent fractions, the key is to change the denominator while maintaining the fraction's value. For example, you may need to convert \( \frac{2}{x} \) to have a denominator of \( 15x \) instead of \( x \).
To change the denominator from \( x \) to \( 15x \), we must multiply it by 15. Remember: whatever you do to the denominator, you must also do to the numerator to keep the fraction equivalent.
This multiplication ensures that the fraction still represents the same quantity or portion of the whole.

Multiplying Factors to Find Equivalent Fractions

Finding equivalent fractions involves using multiplying factors to adjust both the numerator and denominator properly.
To transform \( \frac{2}{x} \) into a fraction with a denominator of \( 15x \), identify what you need to multiply \( x \) by to get \( 15x \).
This multiplying factor is determined by dividing the target denominator by the original denominator: \[ \frac{15x}{x} = 15 \]This means you multiply both the numerator and the denominator of \( \frac{2}{x} \) by 15, resulting in:- New numerator: \( 2 \times 15 = 30 \)- New denominator: \( x \times 15 = 15x \)These calculations give us the equivalent fraction \( \frac{30}{15x} \).
It's important to multiply consistently to maintain the equality and value of the fraction during transformation.