Problem 66 - Prealgebra

Question

Write each fraction as an equivalent fraction with denominator 6. $$\frac{2}{3}$$

Step-by-Step Solution

Answer

The equivalent fraction with denominator 6 is \( \frac{4}{6} \).

Step 1: Understanding the Task

We need to convert the fraction \( \frac{2}{3} \) into an equivalent fraction that has a denominator of 6. This means both fractions represent the same value, but the second has a specified denominator of 6.

Step 2: Finding the Multiplying Factor

To make a denominator of 6 from 3, determine what number you need to multiply 3 by to get 6. The multiplication factor is 2 because \( 3 \times 2 = 6 \).

Step 3: Adjust the Numerator

To maintain equivalence, you must also multiply the numerator by the same factor. Multiply the numerator 2 by the same factor (2) which gives \( 2 \times 2 = 4 \).

Step 4: Construct the Equivalent Fraction

Now, you have the new numerator and denominator. The equivalent fraction of \( \frac{2}{3} \) with a denominator of 6 is \( \frac{4}{6} \).

Key Concepts

Fractions Multiplication

When working with fractions, one essential process is multiplying them, which means dealing with both the numerator and denominator. In our exercise, we start with the fraction \( \frac{2}{3} \) and want to find an equivalent representation with a specific denominator. This requires multiplying both the numerator and the denominator by a strategic number. In basic fraction multiplication, the principle is simple:

• Multiply the numerators of the fractions together.
• Multiply the denominators of the fractions together.
• Simplify the fraction if needed.

However, in our exercise, we only need to apply multiplication to transform the original fraction to an equivalent one with a specified denominator.For example, we identified we need to multiply both the numerator and denominator by 2 to reach our goal denominator of 6. This action scales the fraction without changing its actual value, demonstrating the power of proper multiplication in fractions.

Denominator Adjustment

Adjusting the denominator is a critical skill in fraction manipulation. We often need to change the denominator to compare fractions, add them together, or meet other specific criteria. In our case, we wanted to transform \( \frac{2}{3} \) into a fraction with a denominator of 6.To perform a denominator adjustment effectively:

• Determine the factor needed to convert the old denominator to the new one. For \( \frac{2}{3} \), we multiply \( 3 \) by \( 2 \) to get \( 6 \).
• Ensure you multiply both numerator and denominator by this factor to maintain the fraction's value. This means any transformation must be applied equally to both the top and bottom of the fraction.
• This action ensures the fraction's equivalency stays intact.

Such adjustments allow different fractions to "speak the same language," or in math terms, share common denominators.

Numerator Transformation

Once we've determined the multiplying factor needed to adjust the denominator, we must transform the numerator accordingly. This step is crucial because it maintains the balance of the fraction's value.In our initial problem, where \( \frac{2}{3} \) needs to become \( \frac{x}{6} \):

• We already identified the need for a multiplying factor of \( 2 \).
• We then apply this factor, multiplying the numerator \( 2 \) by \( 2 \) to produce \( 4 \).
• Thus, the adjusted fraction \( \frac{4}{6} \) is equivalent to the original.

This approach, called numerator transformation, ensures the new fraction is not only equivalent structurally, but value-wise as well. Important note: Both numerator and denominator transformations are performed in tandem to ensure the equivalency of the original and adjusted fractions.