Problem 39 - Prealgebra

Question

Write each fraction as an equivalent fraction with denominator \(12 x .\) $$\frac{1}{6}$$

Step-by-Step Solution

Answer

The equivalent fraction is \(\frac{2x}{12x}\).

Step 1: Understanding the Problem

We need to convert the given fraction \(\frac{1}{6}\) into an equivalent fraction with the denominator \(12x\). This means we must find a fraction that is equal in value but has \(12x\) as its denominator.

Step 2: Finding the Multiplicative Factor

To convert the denominator from \(6\) to \(12x\), we must determine what number to multiply \(6\) by to get \(12x\). Solve the equation: \(6 \times k = 12x\). This gives us \(k = \frac{12x}{6} = 2x\).

Step 3: Converting the Fraction

Now, we multiply both the numerator and the denominator of \(\frac{1}{6}\) by \(2x\). This results in: \(\frac{1 \times 2x}{6 \times 2x} = \frac{2x}{12x}\).

Step 4: Final Equivalent Fraction

The fraction \(\frac{1}{6}\) can be rewritten as \(\frac{2x}{12x}\) which is the equivalent fraction with the required denominator of \(12x\).

Key Concepts

Understanding Denominators

In fractions, the denominator plays a crucial role in determining the size of each part of the fraction. The denominator is the number displayed below the line in a fraction. It indicates into how many equal parts the whole is divided.
For example, in the fraction \( \frac{1}{6} \), the denominator is 6, which means the whole is divided into 6 equal parts. Each part is one-sixth of the whole.

• The denominator cannot be zero because dividing something into zero parts is not possible.
• When comparing fractions, knowing the denominators helps in understanding which fractions are larger or smaller.

When converting a fraction to have a different denominator, as in this exercise, our goal is to adjust the fraction so that the new denominator represents the same portion of the whole. This involves using a multiplicative factor that will allow us to change both the numerator and the denominator while maintaining the fraction's value.

Numerator Explained

Just as important as the denominator, the numerator tells us how many parts of the whole we have. It is the number above the line in a fraction.
In the fraction \( \frac{1}{6} \), the numerator is 1, which means we have one part of the six equal parts.

• A numerator of 0 always means the fraction is equal to zero, no matter what the denominator is (except, of course, a denominator of zero, which is undefined).
• When both the numerator and denominator are multiplied by the same factor, the fraction remains equivalent.

In the exercise, to convert \( \frac{1}{6} \) to an equivalent fraction with the denominator \( 12x \), we found a multiplicative factor, \( 2x \). We then multiplied the numerator (1) by this factor to adjust the fraction properly without changing its value.

Role of the Multiplicative Factor

The multiplicative factor is crucial when creating equivalent fractions. It is the number by which both the numerator and the denominator are multiplied in order to find an equivalent fraction with a new denominator.
In this case, when converting \( \frac{1}{6} \) to an equivalent fraction with a denominator of \( 12x \), we needed to determine the appropriate multiplicative factor.

• We solve for the factor using the equation: \( 6 \times k = 12x \), where \( k \) is our multiplicative factor.
• Solving this gives us \( k = \frac{12x}{6} = 2x \), telling us exactly how much to multiply both the numerator and denominator.

By using the multiplicative factor \( 2x \), we ensure the fraction \( \frac{1}{6} \) becomes \( \frac{2x}{12x} \), showing that they are indeed equivalent.