Which Expression Is Equivalent To The Following Complex Fraction?$\[ \frac{1-\frac{1}{x}}{2} \\]A. \[$\frac{x-1}{2x}\$\]B. \[$\frac{-1}{2}\$\]C. \[$\frac{2x-2}{x}\$\]D. \[$\frac{2x}{x-1}\$\]

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Introduction

Complex fractions can be intimidating, but with the right approach, they can be simplified with ease. In this article, we will explore the process of simplifying complex fractions and apply it to a specific problem. We will also examine the different options provided and determine which one is equivalent to the given complex fraction.

Understanding Complex Fractions

A complex fraction is a fraction that contains another fraction in its numerator or denominator. It can be written in the form of abcd\frac{\frac{a}{b}}{\frac{c}{d}} or abc\frac{a}{\frac{b}{c}}. To simplify a complex fraction, we need to multiply the numerator and denominator by the reciprocal of the other fraction.

Simplifying the Given Complex Fraction

The given complex fraction is 1−1x2\frac{1-\frac{1}{x}}{2}. To simplify this fraction, we need to multiply the numerator and denominator by the reciprocal of the fraction in the numerator.

Step 1: Multiply the Numerator and Denominator by the Reciprocal of the Fraction in the Numerator

The reciprocal of the fraction in the numerator is xx. Therefore, we multiply the numerator and denominator by xx.

1−1x2⋅xx=x−1+12x\frac{1-\frac{1}{x}}{2} \cdot \frac{x}{x} = \frac{x - 1 + 1}{2x}

Step 2: Simplify the Numerator

The numerator can be simplified by combining the terms.

x−1+12x=x2x\frac{x - 1 + 1}{2x} = \frac{x}{2x}

Step 3: Cancel Out Common Factors

The numerator and denominator have a common factor of xx. We can cancel out this factor to simplify the fraction further.

x2x=12\frac{x}{2x} = \frac{1}{2}

Examining the Options

Now that we have simplified the complex fraction, let's examine the options provided.

Option A: x−12x\frac{x-1}{2x}

This option is not equivalent to the simplified complex fraction. The numerator and denominator have a common factor of xx, but it is not canceled out.

Option B: −12\frac{-1}{2}

This option is equivalent to the simplified complex fraction. The numerator and denominator have been simplified, and the common factor of xx has been canceled out.

Option C: 2x−2x\frac{2x-2}{x}

This option is not equivalent to the simplified complex fraction. The numerator and denominator have a common factor of xx, but it is not canceled out.

Option D: 2xx−1\frac{2x}{x-1}

This option is not equivalent to the simplified complex fraction. The numerator and denominator have a common factor of xx, but it is not canceled out.

Conclusion

In conclusion, the correct answer is Option B: −12\frac{-1}{2}. This option is equivalent to the simplified complex fraction, and it has been simplified correctly.

Frequently Asked Questions

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains another fraction in its numerator or denominator.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to multiply the numerator and denominator by the reciprocal of the other fraction.

Q: What is the reciprocal of a fraction?

A: The reciprocal of a fraction is obtained by swapping the numerator and denominator.

Final Thoughts

Simplifying complex fractions can be a challenging task, but with the right approach, it can be done with ease. By following the steps outlined in this article, you can simplify complex fractions and determine which option is equivalent to the given complex fraction. Remember to multiply the numerator and denominator by the reciprocal of the other fraction and cancel out common factors to simplify the fraction further.

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Introduction

Complex fractions can be a challenging topic for many students. In our previous article, we explored the process of simplifying complex fractions and applied it to a specific problem. In this article, we will provide a Q&A guide to help you better understand complex fractions and how to simplify them.

Q&A

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains another fraction in its numerator or denominator. It can be written in the form of abcd\frac{\frac{a}{b}}{\frac{c}{d}} or abc\frac{a}{\frac{b}{c}}.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to multiply the numerator and denominator by the reciprocal of the other fraction. The reciprocal of a fraction is obtained by swapping the numerator and denominator.

Q: What is the reciprocal of a fraction?

A: The reciprocal of a fraction is obtained by swapping the numerator and denominator. For example, the reciprocal of ab\frac{a}{b} is ba\frac{b}{a}.

Q: How do I multiply the numerator and denominator by the reciprocal of the other fraction?

A: To multiply the numerator and denominator by the reciprocal of the other fraction, you need to multiply the numerator by the denominator of the other fraction and the denominator by the numerator of the other fraction.

Q: What is the difference between a complex fraction and a simple fraction?

A: A simple fraction is a fraction that does not contain another fraction in its numerator or denominator. A complex fraction, on the other hand, contains another fraction in its numerator or denominator.

Q: Can I simplify a complex fraction by canceling out common factors?

A: Yes, you can simplify a complex fraction by canceling out common factors. However, you need to make sure that the common factors are in the numerator and denominator.

Q: How do I determine which option is equivalent to the given complex fraction?

A: To determine which option is equivalent to the given complex fraction, you need to simplify the complex fraction and compare it with the options provided.

Q: What are some common mistakes to avoid when simplifying complex fractions?

A: Some common mistakes to avoid when simplifying complex fractions include:

  • Not multiplying the numerator and denominator by the reciprocal of the other fraction
  • Not canceling out common factors
  • Not simplifying the numerator and denominator correctly

Examples

Example 1: Simplifying a Complex Fraction

Simplify the complex fraction 1234\frac{\frac{1}{2}}{\frac{3}{4}}.

Solution

To simplify the complex fraction, we need to multiply the numerator and denominator by the reciprocal of the other fraction.

1234â‹…43=46\frac{\frac{1}{2}}{\frac{3}{4}} \cdot \frac{4}{3} = \frac{4}{6}

We can simplify the fraction further by canceling out common factors.

46=23\frac{4}{6} = \frac{2}{3}

Example 2: Determining Which Option is Equivalent to the Given Complex Fraction

Determine which option is equivalent to the complex fraction 1−1x2\frac{1-\frac{1}{x}}{2}.

Options

A. x−12x\frac{x-1}{2x} B. −12\frac{-1}{2} C. 2x−2x\frac{2x-2}{x} D. 2xx−1\frac{2x}{x-1}

Solution

To determine which option is equivalent to the complex fraction, we need to simplify the complex fraction and compare it with the options provided.

1−1x2=x−1+12x=x2x=12\frac{1-\frac{1}{x}}{2} = \frac{x - 1 + 1}{2x} = \frac{x}{2x} = \frac{1}{2}

The correct answer is Option B: −12\frac{-1}{2}.

Conclusion

In conclusion, complex fractions can be a challenging topic, but with the right approach, they can be simplified with ease. By following the steps outlined in this article, you can simplify complex fractions and determine which option is equivalent to the given complex fraction. Remember to multiply the numerator and denominator by the reciprocal of the other fraction and cancel out common factors to simplify the fraction further.

Final Thoughts

Simplifying complex fractions is an important skill that can be applied to a wide range of mathematical problems. By practicing and mastering this skill, you can become more confident and proficient in solving complex mathematical problems.