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Chapter 6: Problem 23

Use either method to simplify each complex fraction. \(\frac{\frac{8 x-24 y}{10}}{\frac{x-3 y}{5 x}}\)

Short Answer

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4x

Step by step solution

01

Identify the complex fraction

The complex fraction given is \ \ \(\frac{\frac{8x-24y}{10}}{\frac{x-3y}{5x}}\). The goal is to simplify this fraction.
02

Simplify the numerator

The numerator of the complex fraction is \(\frac{8x-24y}{10}\). Look for any common factors in the numerator. \ \ Since \(8x - 24y\) can be factored as \(8(x - 3y)\), the expression becomes \ \ \(\frac{8(x - 3y)}{10}\). To simplify, factor out the common terms: \(8(x - 3y)/10 = \frac{4}{5}\cdot(x - 3y)\).
03

Simplify the denominator

The denominator of the complex fraction is \(\frac{x-3y}{5x}\). This cannot be simplified further.
04

Multiply by the reciprocal

A complex fraction \(\frac{A}{B}\) can be simplified by multiplying numerator (A) by the reciprocal of the denominator (B). So, \ \ \(\frac{\frac{4}{5}(x - 3y)}{\frac{x-3y}{5x}} = \frac{4}{5}(x - 3y) \cdot \frac{5x}{x-3y}\).
05

Simplify the expression

The expression \(\frac{4}{5}(x - 3y) \cdot \frac{5x}{x-3y}\) involves multiplying the fractions. The \(x - 3y\) terms in the numerator and the denominator cancel each other out, leaving: \ \ \(\frac{4}{5} \cdot 5x = 4x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Polynomials
Factoring polynomials is a key step when simplifying complex fractions. By breaking down a polynomial into simpler components or 'factors,' you make the expression more manageable. For instance, in the given problem, the expression \(8x - 24y\) can be factored as \(8(x - 3y)\).

Here's a step-by-step guide to factoring polynomials:
  • Identify the greatest common factor (GCF) of the polynomial's terms. The GCF is the largest number or expression that divides each term without a remainder.
  • Factor out the GCF. In our example, \(8\) is the GCF of \(8x\) and \(24y\).
  • Rewrite the polynomial as a product of the GCF and the remaining expression. So, \(8x - 24y\) becomes \(8(x - 3y)\).
Multiplying Fractions
Multiplying fractions is straightforward once you grasp the basic rule: multiply the numerators together and the denominators together. This rule applies whether you're dealing with simple or complex fractions.

Here are the steps to multiply fractions:
  • Write down both fractions.
  • Multiply the numerators (the top numbers) together to get the new numerator.
  • Multiply the denominators (the bottom numbers) together to get the new denominator.
  • Simplify the resulting fraction if possible.
Taking the exercise example, when multiplying \(\frac{4}{5}(x - 3y)\) by \(\frac{5x}{x - 3y}\), you get \(4(x - 3y) \cdot \frac{5x}{x - 3y}\). Notice that the \(x - 3y\) terms will eventually cancel out.
Reciprocal of a Fraction
To simplify complex fractions, you often need to use the reciprocal of a fraction. The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).

Here's how to find and use the reciprocal:
  • Flip the numerator and the denominator of the fraction. For example, the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).
  • To divide by a fraction, multiply by its reciprocal. In the exercise, we simplify the complex fraction by multiplying by the reciprocal of the denominator: \(\frac{\frac{8(x - 3y)}{10}}{\frac{x - 3y}{5x}}\) becomes \(\frac{8(x - 3y)}{10} \cdot \frac{5x}{x - 3y}\). This makes the expression easier to work with.
Canceling Common Factors
Canceling common factors is a useful technique when simplifying fractions. It helps reduce the fraction to its simplest form.

Here's a step-by-step guide to canceling common factors:
  • Identify any identical factors in the numerator and the denominator.
  • If the same factor appears in both places, you can remove it from the fraction.
  • Simplify the remaining fraction.
In our exercise, once we multiply by the reciprocal, we get \(\frac{4}{5}(x - 3y) \cdot \frac{5x}{x - 3y}\). The \(x - 3y\) terms in the numerator and the denominator cancel each other out, leaving \(4x\) as the simplified result. This process eliminates the more complex parts of the fraction, making it simpler to understand and work with.

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Most popular questions from this chapter

Multiply or divide as indicated. $$ \frac{(-3 m n)^{2} \cdot 64\left(m^{2} n\right)^{3}}{16 m^{2} n^{4}\left(m n^{2}\right)^{3}} \div \frac{24\left(m^{2} n^{2}\right)^{4}}{\left(3 m^{2} n^{3}\right)^{2}} $$

Multiply or divide as indicated. $$ \frac{3 x+12}{4 x} \div \frac{4 x+16}{6 x^{2}} $$

Multiply or divide as indicated. $$ \frac{t^{2}-49}{t^{2}+4 t-21} \cdot \frac{t^{2}+8 t+15}{t^{2}-2 t-35} $$

Multiply or divide as indicated. $$ \frac{3 k^{2}+17 k p+10 p^{2}}{6 k^{2}+13 k p-5 p^{2}} \div \frac{6 k^{2}+k p-2 p^{2}}{6 k^{2}-5 k p+p^{2}} $$

As review, multiply or divide the rational numbers as indicated. Write answers in lowest terms. $$ \frac{5}{9} \cdot \frac{12}{25} $$
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