Problem 63 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The equivalent fraction is \( \frac{3}{6} \).

Step 1: Identify the Target Denominator

The problem requires that we write \( \frac{1}{2} \) as an equivalent fraction with a denominator of 6. Thus, the target denominator is 6.

Step 2: Determine the Multiplicative Factor

To find the equivalent fraction, we need to multiply both the numerator and the denominator of \( \frac{1}{2} \) by the same number to obtain a new denominator of 6. We determine that multiplying 2 by 3 will give us 6. Therefore, the multiplicative factor is 3.

Step 3: Multiply Numerator and Denominator

Apply the multiplicative factor (3) to both the numerator and denominator: \( \frac{1 \times 3}{2 \times 3} \). This simplifies to \( \frac{3}{6} \).

Step 4: Verify the Result

We need to check if \( \frac{3}{6} \) is indeed an equivalent fraction to \( \frac{1}{2} \). Simplifying \( \frac{3}{6} \) by dividing the numerator and denominator by their greatest common divisor (3) gives us \( \frac{1}{2} \). This confirms that \( \frac{3}{6} \) is correct.

Key Concepts

Understanding Fractions

Fractions are a fundamental part of mathematics, representing parts of a whole. They consist of two main parts: a numerator and a denominator. The numerator is the number above the fraction line and indicates how many parts of the whole are being considered. The denominator, on the other hand, is below the line and tells into how many equal parts the whole is divided.

Visualization can be helpful. Imagine a pizza sliced into 4 pieces: if you have 1 slice, the fraction representing your share is \( \frac{1}{4} \). Hence, fractions are crucial for depicting proportional relationships and quantities less than a whole number.

Fractions can also express equal distributions, such as sharing a cake equally among friends. Understanding fractions allows us to compare, add, subtract, multiply, and divide parts of a whole in various situations.

What Are Denominators?

The denominator in a fraction serves as the foundation. It indicates the total number of equal parts in a whole. In a fraction like \( \frac{1}{2} \), the '2' denotes that the whole is split into two parts. A different denominator means a different size for each part. For instance, with a denominator of 6, each part is one-sixth of the whole.

Denominators are important when comparing fractions. They must be the same if you want to perform arithmetic operations such as addition or subtraction. By converting fractions to equivalent forms with a common denominator, these operations become much more straightforward.

When adjusting denominators, remember that the whole amount the fraction represents remains unchanged; you're only changing the way it's divided. This concept is fundamental when learning about equivalent fractions.

Numerator and Denominator Multiplication for Equivalent Fractions

Creating equivalent fractions involves multiplying both the numerator and the denominator by the same number. This operation doesn't change the value of the fraction because you are essentially multiplying by 1. For instance, changing \( \frac{1}{2} \) to have a denominator of 6 involves multiplying both parts by 3: \( \frac{1 \times 3}{2 \times 3} = \frac{3}{6} \).

The essence of this process is that the ratio between the numerator and the denominator remains the same, hence the fraction's value is unchanged. This method is powerful for simplifying, comparing, and adding fractions since it allows us to adjust fractions without changing their value.

Once you've obtained an equivalent fraction with a common denominator, it's easy to handle operations like addition or subtraction since you're working with comparable parts. Practicing these multiplications will deepen your understanding and comfort with equivalent fractions.