Problem 54 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{\frac{3}{w}-\frac{6}{2 w+1}}{\frac{5}{w}+\frac{8}{2 w+1}}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3}{18w+5} \).

Step 1: Identify and Simplify Each Fraction

Before addressing the entire complex fraction, we observe the individual fractions in the numerator and denominator: \( \frac{3}{w} \), \( \frac{6}{2w+1} \), \( \frac{5}{w} \), and \( \frac{8}{2w+1} \). Since they involve different denominators \( w \) and \( 2w+1 \), they cannot be directly added or subtracted without a common denominator. Here, we note that we will need the least common denominator (LCD) for each addition or subtraction.

Step 2: Find the Least Common Denominator (LCD)

The least common denominator for \( w \) and \( 2w+1 \) is their product \( w(2w+1) \). We will use this to rewrite and combine the terms in both the numerator and the denominator.

Step 3: Rewrite and Simplify the Numerator

Rewrite \( \frac{3}{w} \) and \( \frac{6}{2w+1} \) using the least common denominator \( w(2w+1) \):\[ \frac{3}{w} = \frac{3(2w+1)}{w(2w+1)} = \frac{6w + 3}{w(2w+1)} \]\[ \frac{6}{2w+1} = \frac{6w}{w(2w+1)} \]Combine these to get the new numerator:\[ \frac{6w + 3 - 6w}{w(2w+1)} = \frac{3}{w(2w+1)} \]

Step 4: Rewrite and Simplify the Denominator

Rewrite \( \frac{5}{w} \) and \( \frac{8}{2w+1} \) using the least common denominator \( w(2w+1) \):\[ \frac{5}{w} = \frac{5(2w+1)}{w(2w+1)} = \frac{10w + 5}{w(2w+1)} \]\[ \frac{8}{2w+1} = \frac{8w}{w(2w+1)} \]Combine these to get the new denominator:\[ \frac{10w + 5 + 8w}{w(2w+1)} = \frac{18w + 5}{w(2w+1)} \]

Step 5: Combine and Simplify Complex Fraction

Now that the numerator and the denominator each have the same denominator \( w(2w+1) \), they can be treated as a single fraction. Simplify the complex fraction:\[ \frac{ \frac{3}{w(2w+1)} }{ \frac{18w+5}{w(2w+1)} } = \frac{3}{18w+5} \]

Step 6: Simplify if Possible

The final expression \( \frac{3}{18w+5} \) cannot be simplified further, as there are no common factors between the numerator and the denominator.

Key Concepts

Complex Fractions

A complex fraction is essentially a fraction where the numerator, the denominator, or sometimes both, are themselves fractions. This can look a bit tricky at first, but once you break it down, it's quite manageable. Consider something like this:
\[\frac{\frac{A}{B}}{\frac{C}{D}}\]This structure can seem overwhelming, but always remember that it is simply a fraction divided by another fraction. Sounds simpler already, right? To solve such a fraction, you need to address the inner basic fractions first.

• Identify the individual fractions in the numerator and the denominator.
• Find their Least Common Denominator (LCD), which we'll explore more in the next section.
• Convert each fraction accordingly to have a common base.
• Finally, divide the simplified numerator by the simplified denominator, and voilà, your complex fraction is simplified.

Once you handle these incremental steps, complex fractions are not such a complicated monster after all! It just takes a bit of practice to become familiar with juggling multiple fractions.

Least Common Denominator

The Least Common Denominator, or LCD, is crucial when dealing with fractions, especially in operations such as addition or subtraction. Whenever you need to combine fractions, they should have the same denominator, called the common denominator. For two fractions like \(\frac{a}{b}\) and \(\frac{c}{d}\), the LCD is the smallest value that both denominators can divide without leaving a remainder.
In our exercise, for instance, we look at denominators like \(w\) and \(2w+1\). The LCD for these is simply their product: \(w(2w+1)\). Here’s why finding the LCD is important:

• Allows you to rewrite fractions so that they have the same base.
• Makes addition or subtraction straightforward without creating fractional chaos.
• Simplifies the problem, making it easier to solve.

Once you rewrite each fraction using the LCD, you proceed by combining them, just like adding regular numbers. Mastering the use of LCD will streamline your work with fractions and complex fractions alike.

Simplification Steps

Simplification is all about making an expression as simple as possible without changing its value. After aligning denominators in a complex fraction using the Least Common Denominator (LCD), the next crucial step is simplification.
Here's a coherent way to tackle simplification:

• Begin by ensuring both the numerator and the denominator of your complex expression share the same denominator.
• Combine terms in the numerator and the denominator as needed. This might mean straightforward addition or subtraction, as guided by your newly agreed-upon denominators.
• Simplify each resulting term and ensure you're left with a single fraction to work with.
• Remember to check if further simplification is possible, like reducing the fraction by dividing common terms in the numerator and denominator.

If the math turns out clean, your expression is at its simplest form. If not, go back and meticulously reassess each step. Simplification not only leads to cleaner results but also to greater understanding in solving fractions.