Problem 32 - Prealgebra

Question

Write each of the following fractions as an equivalent fraction with denominator 6. $$\frac{1}{2}$$

Step-by-Step Solution

Answer

\( \frac{1}{2} \) as a fraction with denominator 6 is \( \frac{3}{6} \).

Step 1: Understanding the Objective

We are given a fraction \( \frac{1}{2} \) and need to express it as an equivalent fraction with a new denominator of 6.

Step 2: Determine the Factor Needed for the Denominator

We need to modify the denominator of \( \frac{1}{2} \) to become 6. The current denominator is 2, so we calculate the factor needed to multiply by 2 to get 6, which is 3, since \( 2 \times 3 = 6 \).

Step 3: Adjust the Numerator

In order to maintain the value of the fraction while changing the denominator, we also multiply the numerator by 3, the same factor we used for the denominator. Thus, \( 1 \times 3 = 3 \).

Step 4: Write the Equivalent Fraction

We now use the adjusted numbers to write the new fraction: \( \frac{3}{6} \). This is the equivalent fraction of \( \frac{1}{2} \) with a denominator of 6.

Key Concepts

Denominator Adjustment

Making adjustments to a fraction's denominator is an essential skill. It allows us to represent the same value in different forms. Here’s how it works:

To find a common denominator, you need to decide what number the denominator should become. In some problems, this desired number is given (like 6 in this problem). The key is to identify what you must multiply the current denominator by to reach that new target number.

For example, with the fraction \(\frac{1}{2}\), we wanted the denominator to be 6. Since the original denominator is 2, and you multiply 2 by 3 to get 6 (i.e., \(2 \times 3 = 6\)), 3 is the factor you need. Remember, consistency is crucial. You must apply this factor to both parts of the fraction to preserve its value.

Numerator Adjustment

Once you've found the factor needed for the denominator adjustment, it’s time to apply the same factor to the numerator. This process ensures the fraction remains equal to its original value.

Take the example of \(\frac{1}{2}\):

• The denominator adjusted with the factor 3 becomes 6. That step is clear.
• Now, apply that factor of 3 to the numerator as well.
• So, multiplying the numerator 1 by 3 gives \(1 \times 3 = 3\). This results in a new numerator.

This step of adjusting the numerator helps to maintain the overall value of the fraction, and thus, \(\frac{3}{6}\) is an equivalent representation of \(\frac{1}{2}\).

Fraction Equivalence

Understanding equivalent fractions is all about seeing how numbers relate proportionally. Equivalent fractions are different fractions that represent the same part of a whole.

For instance, \(\frac{1}{2}\) and \(\frac{3}{6}\) might look different, but they are equivalent. Here's why:

• Both fractions occupy the same space on a number line.
• Both represent half of something when simplified to their lowest terms.
• The process of adjusting numerators and denominators, using the same factor, shows the multiplicative identity principle, i.e., multiplying by 1 (or \(3/3\), which equals 1) does not change the value of a fraction.

Recognizing and using equivalent fractions simplifies addition, subtraction, and comparison of fractions in mathematical problems.