Problem 59 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{\frac{10 x}{x-3}}{\frac{6}{x-3}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{5x}{3} \).

Step 1: Identify the Complex Fraction

The given expression is a complex fraction: \( \frac{\frac{10x}{x-3}}{\frac{6}{x-3}} \). A complex fraction is a fraction where the numerator and/or the denominator is also a fraction.

Step 2: Apply Division Rule for Fractions

To simplify a complex fraction, recall that dividing by a fraction is the same as multiplying by its reciprocal. This means \( \frac{\frac{10x}{x-3}}{\frac{6}{x-3}} \) is equivalent to multiplying \( \frac{10x}{x-3} \) by the reciprocal of \( \frac{6}{x-3} \), which is \( \frac{x-3}{6} \).

Step 3: Perform the Multiplication

Multiply the fractions: \( \frac{10x}{x-3} \times \frac{x-3}{6} \). When multiplying fractions, multiply the numerators together and the denominators together: \( \frac{10x \times (x-3)}{6 \times (x-3)} \).

Step 4: Simplify the Expression

Notice that \( x-3 \) in the numerator and denominator cancels out, leaving \( \frac{10x}{6} \).

Step 5: Reduce the Fraction

Simplify \( \frac{10x}{6} \) by dividing both the numerator and the denominator by their greatest common divisor, 2: \( \frac{5x}{3} \).

Key Concepts

Fraction Simplification

Fraction simplification is a foundational skill in algebra that helps make expressions easier to work with. To simplify a fraction, the goal is to reduce it to its simplest form by eliminating any common factors shared by the numerator and the denominator.
To do this:

• Identify common factors of the numerator and the denominator.
• Divide both the numerator and the denominator by these common factors until no further simplification is possible.

In the exercise, after applying all steps, we identified that both 10x and 6 were divisible by 2. By dividing the numerator and denominator by this common factor, we finalised the simplification, resulting in the simpler fraction \( \frac{5x}{3} \). This approach helps ensure clarity and prepares the expression for any further operations.

Division of Fractions

Dividing fractions might seem tricky at first, but with the right approach, it becomes much simpler. The main "trick" lies in converting a division problem into a multiplication problem.
Here's how:

• When you divide by a fraction, it is equivalent to multiplying by its reciprocal.
• The reciprocal of a fraction is simply flipping its numerator and denominator.

In our exercise, the complex fraction \( \frac{\frac{10x}{x-3}}{\frac{6}{x-3}} \) is converted to \( \frac{10x}{x-3} \times \frac{x-3}{6} \). Instead of dividing by \( \frac{6}{x-3} \), we multiply by its reciprocal, \( \frac{x-3}{6} \). By changing our perspective from division to multiplication, the process becomes straightforward.

Reciprocals in Algebra

Reciprocals play a crucial role in simplifying complex fractions and performing division operations. The reciprocal of a number or expression is essentially what you would multiply the original number by to get one.
It's calculated by swapping the numerator and the denominator. Some key points to remember about reciprocals:

• The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \).
• Multiplying a fraction by its reciprocal yields 1, which is the fundamental property used to "cancel out" fractions.

In the given exercise, we focused on finding the reciprocal of the denominator fraction \( \frac{6}{x-3} \). By doing so, we transformed a complex division into a straightforward multiplication, leading us to an easier way to simplify the expression. Understanding and applying the concept of reciprocals in algebra helps to simplify operations and solve problems efficiently.