Problem 31 - Prealgebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to rewrite the fraction \(\frac{2}{3}\) as an equivalent fraction that has a denominator of 6. Equivalent fractions are fractions that represent the same value, even though they may have different numerators and denominators.

Step 2: Determine the Multiplication Factor

To find an equivalent fraction with a denominator of 6, we need to determine what number to multiply 3 by to get 6. We find that multiplying 3 by 2 equals 6. Therefore, we will multiply both the numerator and the denominator by 2 to maintain the equality.

Step 3: Multiply Numerator and Denominator

Since multiplying the denominator 3 by 2 gives us 6, we also multiply the numerator 2 by 2. This gives us 4 in the numerator. Therefore, when we multiply both parts of the fraction \(\frac{2}{3}\) by 2, we get \(\frac{2 \times 2}{3 \times 2} = \frac{4}{6}\).

Step 4: Confirm Equivalent Fraction

Check the equivalency by simplifying \(\frac{4}{6}\) back to \(\frac{2}{3}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. Dividing gives us \(\frac{4 \div 2}{6 \div 2} = \frac{2}{3}\), thus confirming the equivalency.

Key Concepts

Fraction Multiplication

Multiplying fractions involves multiplying both numerators with each other and denominators together. It's straightforward and follows the basic rule: numerator times numerator and denominator times denominator.
For example, to express \(\frac{2}{3}\) as an equivalent fraction with a denominator of 6, we multiply both its numerator and denominator by 2.
This involves simple multiplication:

• Numerator: \(2 \times 2 = 4\)
• Denominator: \(3 \times 2 = 6\)

Hence, \(\frac{2}{3}\) becomes \(\frac{4}{6}\). This method of scaling both the top and the bottom keeps the fraction equivalent because fractions essentially represent division.

Common Denominators

Finding a common denominator allows you to compare or add fractions more easily. In simple terms, fractions need to have the same bottom numbers when performing these operations.
To convert \(\frac{2}{3}\) to have a common denominator of 6, first decide what number must be multiplied by the current denominator, which in this case is 3, to reach 6.
Multiplication by 2 achieves this. Therefore, multiplying both the numerator and the denominator by this number creates a new fraction \(\frac{4}{6}\) that has the desired denominator of 6.

Simplifying Fractions

Simplifying, or reducing, a fraction means rewriting it in its most compact form, where the top and bottom numbers are smaller but the value remains unchanged.
This is done by finding the greatest common divisor (GCD) of the numerator and the denominator. For \(\frac{4}{6}\), both 4 and 6 can be divided by 2, their GCD.
When you simplify \(\frac{4}{6}\) by dividing both by 2, it returns to \(\frac{2}{3}\), thus confirming that the new fraction truly is equivalent to the original form. Remember, simplifying makes fractions easier to use and understand without altering their value.