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Chapter 6: Problem 12

Use either method to simplify each complex fraction. \(\frac{\frac{24}{t+4}}{\frac{6}{t}}\)

Short Answer

Expert verified
The simplified form is \(\frac{4t}{t+4}\).

Step by step solution

01

Rewrite the complex fraction

Rewrite the division of fractions as a multiplication by the reciprocal. The expression \(\frac{\frac{24}{t+4}}{\frac{6}{t}}\) can be rewritten as \(\frac{24}{t+4} \times \frac{t}{6}\).
02

Multiply the fractions

Multiply the numerators together and the denominators together: \(\frac{24 \times t}{(t+4) \times 6}\).
03

Simplify the expression

Simplify the fraction by canceling common factors in the numerator and the denominator. Both 24 and 6 can be divided by 6: \(\frac{24t}{6(t+4)} = \frac{4t}{t+4}\).

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Key Concepts

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

Complex Fractions
Complex fractions are fractions where either the numerator, the denominator, or both are also fractions. They might look a bit scary, but they aren鈥檛 too difficult to simplify once you know the steps. Consider the complex fraction \(\frac{\frac{24}{t+4}}{\frac{6}{t}}\). Here, we have fractions in both the numerator and denominator. Our goal is to simplify it so it looks like a regular fraction.
Reciprocal
The reciprocal of a fraction is what you get when you swap its numerator and denominator. If you have a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). In the context of simplifying complex fractions, using the reciprocal helps us turn division into multiplication. For instance, \(\frac{\frac{24}{t+4}}{\frac{6}{t}}\) can be rewritten by multiplying with the reciprocal of the denominator. So, it becomes \(\frac{24}{t+4} \times \frac{t}{6}\).
Multiplying Fractions
Once the complex fraction is rewritten to look like \(\frac{24}{t+4} \times \frac{t}{6}\), we can multiply the fractions. To do this, multiply the numerators together and the denominators together. Here, it means we perform the multiplication: \(\frac{24 \times t}{(t+4) \times 6}\). This operation includes the numerator 24 \(t\) and the denominator \(6(t + 4)\).
Simplifying Expressions
After multiplying the fractions, we often need to simplify the resulting fraction. Simplifying means reducing the fraction to its simplest form. We do this by canceling out any common factors in the numerator and the denominator. For \(\frac{24t}{6(t+4)}\), both 24 and 6 can be divided by 6. So, \(\frac{24t}{6(t+4)} = \frac{4t}{t+4}\), which is our final, simplified expression. Simplifying helps make fractions easier to work with and understand.

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Most popular questions from this chapter

Write each rational expression in lowest terms. $$ \frac{7-b}{b-7} $$

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

Write each rational expression in lowest terms. $$ \frac{4 x-9}{4} $$

Which rational expression can be simplified? A. \(\frac{x^{2}+2}{x^{2}}\) B. \(\frac{x^{2}+2}{2}\) C. \(\frac{x^{2}+y^{2}}{y^{2}}\) D. \(\frac{x^{2}-5 x}{x}\)

Multiply or divide as indicated. $$ \frac{(x+2)(x+1)}{(x+3)(x-2)} \cdot \frac{(x+3)(x+4)}{(x+2)(x+1)} $$
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