Problem 41 - Elementary Algebra

Question

For Problems 41-60, simplify each of the complex fractions. $$ \frac{\frac{1}{2}-\frac{3}{4}}{\frac{1}{6}+\frac{1}{3}} $$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( \frac{-1}{2} \).

Step 1: Simplify the Numerator

First, let's simplify the numerator, which is \( \frac{1}{2} - \frac{3}{4} \). To subtract these fractions, we need a common denominator. The least common denominator of 2 and 4 is 4. We rewrite \( \frac{1}{2} \) as \( \frac{2}{4} \). Now we have: \( \frac{2}{4} - \frac{3}{4} = \frac{-1}{4} \).

Step 2: Simplify the Denominator

Next, simplify the denominator, \( \frac{1}{6} + \frac{1}{3} \). The least common denominator of 6 and 3 is 6. Rewrite \( \frac{1}{3} \) as \( \frac{2}{6} \). Add the fractions: \( \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2} \) after simplification.

Step 3: Divide the Numerator by the Denominator

Now, divide the simplified numerator \( \frac{-1}{4} \) by the simplified denominator \( \frac{1}{2} \). This is equivalent to multiplying \( \frac{-1}{4} \) by the reciprocal of \( \frac{1}{2} \), which is \( 2 \). Therefore, \( \frac{-1}{4} \times 2 = \frac{-2}{4} = \frac{-1}{2} \) after simplification.

Key Concepts

Understanding the Least Common Denominator

When we work with fractions, especially while adding or subtracting them, having a common denominator can make the process a whole lot simpler. The least common denominator (LCD) is the smallest number that can be used as a denominator for all fractions in the expression. It's like finding a common language for different fractions so they can "talk" to each other fluently.

• First, identify the denominators of the fractions involved. In our problem, they are 2 and 4 for the numerator, and 6 and 3 for the denominator.
• Next, find the smallest multiple they both share. For 2 and 4, it's 4. For 6 and 3, it's 6.
• Rewrite each fraction using this common denominator. For instance, in the numerator, rewrite \(\frac{1}{2}\) as \(\frac{2}{4}\) so it can be directly subtracted from \(\frac{3}{4}\).

This step makes subtraction or addition more straightforward and helps avoid errors when handling fractions.

Mastering Fraction Subtraction

Subtracting fractions might seem tricky at first, but with the right steps, it becomes straightforward. The key is to ensure both fractions have the same denominator before performing the operation. This is where our least common denominator shines.

• Once both fractions share the same denominator, you simply subtract the numerators and keep the common denominator.
• In our example, \(\frac{1}{2}\) becomes \(\frac{2}{4}\), allowing us to subtract it from \(\frac{3}{4}\)
• The operation looks like this: \(\frac{2}{4} - \frac{3}{4} = \frac{-1}{4}\).

Remember, the denominator remains unchanged. It's essentially the "base" keeping the fractions aligned, even as their values change in the subtraction process. This ensures clarity and precision whenever you face fraction subtraction.

The Magic of Reciprocal Multiplication

Reciprocal multiplication is an elegant solution when dividing fractions. The reciprocal of a number or fraction flips the numerator and the denominator, creating a "mirror image." When you need to divide one fraction by another, multiplying by the reciprocal makes it much simpler.

• Identify the fraction that you need to divide by — in our solution, this is \(\frac{1}{2}\).
• Compute the reciprocal of this fraction, which is \(2\).
• Then, instead of dividing \(\frac{-1}{4}\) by \(\frac{1}{2}\), multiply \(\frac{-1}{4}\) by \(2\).
• This looks like: \(\frac{-1}{4} \times 2 = \frac{-2}{4} = \frac{-1}{2}\).

This technique is powerful because multiplication is often simpler than division. Reciprocal multiplication transforms a division problem into a multiplication one, ensuring results are swiftly and correctly achieved.