Problem 81 - Algebra A Combined Function

Question

Simplify. $$ \frac{\frac{x}{6}}{\frac{2 x}{3}+\frac{1}{2}} $$

Step-by-Step Solution

Answer

\(\frac{x}{4x + 3}\)

Step 1: Simplify the Expression

Let's first write the given expression in a simpler form: \[\frac{\frac{x}{6}}{\frac{2x}{3} + \frac{1}{2}}\]Our aim is to simplify this complex fraction. A good strategy is to work on simplifying the numerator and the denominator separately.

Step 2: Simplify the Numerator

The numerator is just \(\frac{x}{6}\). There is nothing further to simplify here as it is already a simple fraction.

Step 3: Simplify the Denominator

The denominator is \(\frac{2x}{3} + \frac{1}{2}\). To add these two fractions, we need to find a common denominator. The least common multiple of 3 and 2 is 6.

Step 4: Rewrite the Denominator

Convert each term to have a denominator of 6: \[\frac{2x}{3} = \frac{4x}{6} \quad \text{and} \quad \frac{1}{2} = \frac{3}{6}\] Now, rewrite the expression as: \[\frac{4x}{6} + \frac{3}{6} = \frac{4x + 3}{6}\] The denominator is now a single fraction.

Step 5: Rewrite the Entire Expression

Substitute the simplified denominator back into the main expression:\[\frac{\frac{x}{6}}{\frac{4x + 3}{6}}\]Now we have a complex fraction where both numerator and denominator have the same denominator of 6.

Step 6: Simplify the Complex Fraction

Since both the numerator and the denominator of the complex fraction have a common denominator of 6, they cancel out:\[\frac{x}{4x + 3}\] So the simplified form of the expression is \(\frac{x}{4x + 3}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is a fundamental skill in algebra. The goal is to make fractions easier to work with by reducing them to their simplest form.
This involves dividing the numerator and the denominator by their greatest common divisor (GCD).

• Start by determining if there are any common factors in the numerator and the denominator.
• Divide both parts by the greatest common divisor to simplify the fraction.
• Always check your results to ensure the fraction cannot be reduced further.

A simplified fraction is easier to interpret and use in further calculations. Remember, simplifying doesn't change the value of the fraction; it just makes it shorter and less complicated.

Complex Fractions

A complex fraction is one where either the numerator, the denominator, or both are themselves fractions.
These fractions can appear intimidating at first but become manageable once you understand the hierarchy of operations. To simplify a complex fraction, follow these steps:

• Simplify the numerator and the denominator separately if necessary.
• If both are already simple, find a common denominator to combine them if needed.
• Rewrite the complex fraction as a division operation.
• Use the fact that dividing by a fraction is equivalent to multiplying by its reciprocal.

Once this is completed, the complex fraction becomes much easier to work with and solve.

Common Denominator

Finding a common denominator is crucial when adding or subtracting fractions. It allows you to combine the fractions easily by ensuring that they have the same denominator.
Here’s how you find it:

• Identify the denominators of each fraction in your problem.
• Calculate the least common multiple (LCM) of these denominators. This is your common denominator.
• Rewrite each fraction so that it has the identical denominator of the LCM.

Using a common denominator enables you to add or subtract the fractions by keeping the denominator the same, while only performing operations on the numerators. This simplifies fractions, making it easier to solve algebraic problems involving fractions.