Problem 6 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{\frac{6 x-3}{5 x^{2}}}{\frac{2 x-1}{10 x}} $$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( \frac{6}{x} \).

Step 1: Understand the Structure of the Complex Fraction

The given expression is \( \frac{\frac{6x-3}{5x^2}}{\frac{2x-1}{10x}} \). This is a complex fraction because it involves fractions within a fraction. Our task is to simplify this expression to a single fraction.

Step 2: Invert the Denominator and Multiply

To simplify the complex fraction, recall that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, rewrite the expression as \( \frac{6x-3}{5x^2} \times \frac{10x}{2x-1} \).

Step 3: Simplify the Numerator and Denominator

Now, simplify the fractions before carrying out the multiplication. For the numerator fraction \( \frac{6x-3}{5x^2} \), factor out the common factor: \( \frac{3(2x-1)}{5x^2} \). Keep the second fraction as is: \( \frac{10x}{2x-1} \).

Step 4: Carry out the Multiplication

Multiply the fractions: \( \frac{3(2x-1)}{5x^2} \times \frac{10x}{2x-1} \). This becomes \( \frac{3\cdot 10x (2x-1)}{5x^2(2x-1)} \).

Step 5: Cancel Common Factors

Notice that \( (2x-1) \) appears in both the numerator and the denominator and can be canceled out (provided \( 2x-1 eq 0 \)). Now, you're left with \( \frac{30x}{5x^2} \). Simplify further by canceling any common factors: \( \frac{30}{5x} = \frac{6}{x} \).

Step 6: Verify the Simplification

The expression \( \frac{6}{x} \) is the simplest form because no further common factors exist in the numerator and denominator. Therefore, the complex fraction has been fully simplified.

Key Concepts

Simplifying Fractions

When you simplify fractions, it means reducing the fraction to its simplest form. Simplifying involves finding a common factor for both the numerator and the denominator. Once found, this factor is divided out from both to make the fraction smaller and easier to handle. The basic rules are:

• The numerator is the top part of the fraction, and the denominator is the bottom part.
• Identify any common factors between the numerator and the denominator.
• Divide both the numerator and the denominator by their greatest common factor (GCF).

For example, in the fraction \( \frac{36}{48} \), the GCF is 12. Dividing both numbers by 12 simplifies the fraction to \( \frac{3}{4} \). Similarly, in complex fractions, you simplify by looking for common factors and reducing them, as was done in the solved exercise by canceling out \((2x-1)\) from both parts.

Factoring Expressions

Factoring expressions is breaking them down into multiplication like terms. It's like reverse multiplication. By finding the factors, you can easily simplify complex expressions or solve equations. Start by identifying the greatest common factor in all terms. This might be numbers or variables. - Look for numbers that divide each of them evenly.- With variables, see if there is a common one you can take out.For example, in the expression \( 6x - 3 \), you can factor out a 3. This process turns it into \( 3(2x-1) \). Notice how this simplification was fundamental in the exercise provided, allowing the expression to be reduced step by step masterfully.

Reciprocal of Fractions

The reciprocal is a simple yet powerful tool in algebra. It's what you multiply the fraction by to get the number 1. You achieve it by swapping the numerator and the denominator. For example, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).In solving complex fractions, the most significant step involves using the reciprocal to simplify by multiplication. This means instead of dividing by a fraction, you multiply by its reciprocal, leading to a more straightforward single fraction. For instance, in the solved exercise, \( \frac{10x}{2x-1} \) was multiplied in place of dividing by \( \frac{2x-1}{10x} \). This simple trick makes the complexities vanish, one step at a time.Always remember: multiplying by the reciprocal instead of dividing can make your math much tidier!