Which Expression Is Equivalent To The Following Complex Fraction?$\[ \frac{\frac{-2}{x}+\frac{5}{y}}{\frac{3}{y}-\frac{2}{x}} \\]A. \[$\frac{-2y+5x}{3x-2y}\$\]B. \[$\frac{3x-2y}{-2y+5x}\$\]C.

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Understanding Complex Fractions

Complex fractions are mathematical expressions that contain fractions within other fractions. They can be challenging to simplify, but with a clear understanding of the concept and a step-by-step approach, you can master the art of simplifying complex fractions.

The Given Complex Fraction

The given complex fraction is ${ \frac{\frac{-2}{x}+\frac{5}{y}}{\frac{3}{y}-\frac{2}{x}} }$. To simplify this expression, we need to find an equivalent expression that is easier to work with.

Step 1: Find a Common Denominator

To add or subtract fractions, we need to have a common denominator. In this case, the common denominator is xyxy. We can rewrite each fraction with the common denominator:

{ \frac{-2}{x} = \frac{-2y}{xy} \}

{ \frac{5}{y} = \frac{5x}{xy} \}

{ \frac{3}{y} = \frac{3x}{xy} \}

{ \frac{2}{x} = \frac{2y}{xy} \}

Step 2: Add and Subtract the Fractions

Now that we have a common denominator, we can add and subtract the fractions:

{ \frac{-2y}{xy} + \frac{5x}{xy} = \frac{-2y + 5x}{xy} \}

{ \frac{3x}{xy} - \frac{2y}{xy} = \frac{3x - 2y}{xy} \}

Step 3: Simplify the Complex Fraction

Now that we have simplified the numerator and denominator, we can rewrite the complex fraction:

{ \frac{\frac{-2y + 5x}{xy}}{\frac{3x - 2y}{xy}} \}

Step 4: Cancel Out the Common Factors

We can cancel out the common factors in the numerator and denominator:

{ \frac{-2y + 5x}{3x - 2y} \}

Conclusion

The simplified expression is ${ \frac{-2y + 5x}{3x - 2y} }$. This is the equivalent expression to the given complex fraction.

Answer Options

Let's compare the simplified expression with the answer options:

A. ${ \frac{-2y + 5x}{3x - 2y} }$

B. ${ \frac{3x - 2y}{-2y + 5x} }$

C. ${ \frac{3x - 2y}{-2y + 5x} }$

The correct answer is A. ${ \frac{-2y + 5x}{3x - 2y} }$.

Discussion

Simplifying complex fractions requires a clear understanding of the concept and a step-by-step approach. By following the steps outlined in this article, you can master the art of simplifying complex fractions and become more confident in your mathematical abilities.

Tips and Tricks

  • Always start by finding a common denominator.
  • Simplify the numerator and denominator separately.
  • Cancel out common factors in the numerator and denominator.
  • Check your answer by plugging in values for the variables.

Real-World Applications

Simplifying complex fractions has many real-world applications, including:

  • Finance: Simplifying complex fractions can help you understand financial concepts, such as interest rates and investment returns.
  • Science: Simplifying complex fractions can help you understand scientific concepts, such as velocity and acceleration.
  • Engineering: Simplifying complex fractions can help you understand engineering concepts, such as stress and strain.

Conclusion

Simplifying complex fractions is a crucial skill that can help you become more confident in your mathematical abilities. By following the steps outlined in this article, you can master the art of simplifying complex fractions and apply it to real-world problems.

Frequently Asked Questions

Simplifying complex fractions can be a challenging task, but with the right guidance, you can master the art of simplifying complex fractions. Here are some frequently asked questions and answers to help you better understand the concept.

Q: What is a complex fraction?

A: A complex fraction is a mathematical expression that contains fractions within other fractions. It can be a challenging task to simplify complex fractions, but with a clear understanding of the concept and a step-by-step approach, you can master the art of simplifying complex fractions.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to follow these steps:

  1. Find a common denominator.
  2. Simplify the numerator and denominator separately.
  3. Cancel out common factors in the numerator and denominator.
  4. Check your answer by plugging in values for the variables.

Q: What is the common denominator?

A: The common denominator is the least common multiple (LCM) of the denominators of the fractions. It is the smallest number that both denominators can divide into evenly.

Q: How do I find the common denominator?

A: To find the common denominator, you need to list the multiples of each denominator and find the smallest number that appears in both lists.

Q: Can I simplify a complex fraction with different denominators?

A: Yes, you can simplify a complex fraction with different denominators. However, you need to find a common denominator first.

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

A: A simple fraction is a fraction with a single numerator and denominator, while a complex fraction is a fraction with fractions within other fractions.

Q: Can I simplify a complex fraction with negative numbers?

A: Yes, you can simplify a complex fraction with negative numbers. However, you need to follow the same steps as simplifying a complex fraction with positive numbers.

Q: Can I simplify a complex fraction with variables?

A: Yes, you can simplify a complex fraction with variables. However, you need to follow the same steps as simplifying a complex fraction with numbers.

Q: What is the final answer to the given complex fraction?

A: The final answer to the given complex fraction is ${ \frac{-2y + 5x}{3x - 2y} }$.

Q: Can I use a calculator to simplify a complex fraction?

A: Yes, you can use a calculator to simplify a complex fraction. However, it is always best to simplify complex fractions by hand to ensure accuracy.

Q: Can I simplify a complex fraction with multiple variables?

A: Yes, you can simplify a complex fraction with multiple variables. However, you need to follow the same steps as simplifying a complex fraction with a single variable.

Q: Can I simplify a complex fraction with a variable in the denominator?

A: Yes, you can simplify a complex fraction with a variable in the denominator. However, you need to follow the same steps as simplifying a complex fraction with a number in the denominator.

Conclusion

Simplifying complex fractions can be a challenging task, but with the right guidance, you can master the art of simplifying complex fractions. By following the steps outlined in this article and answering the frequently asked questions, you can become more confident in your mathematical abilities and apply it to real-world problems.

Tips and Tricks

  • Always start by finding a common denominator.
  • Simplify the numerator and denominator separately.
  • Cancel out common factors in the numerator and denominator.
  • Check your answer by plugging in values for the variables.
  • Use a calculator to simplify complex fractions, but always double-check your answer by hand.

Real-World Applications

Simplifying complex fractions has many real-world applications, including:

  • Finance: Simplifying complex fractions can help you understand financial concepts, such as interest rates and investment returns.
  • Science: Simplifying complex fractions can help you understand scientific concepts, such as velocity and acceleration.
  • Engineering: Simplifying complex fractions can help you understand engineering concepts, such as stress and strain.

Conclusion

Simplifying complex fractions is a crucial skill that can help you become more confident in your mathematical abilities. By following the steps outlined in this article and answering the frequently asked questions, you can master the art of simplifying complex fractions and apply it to real-world problems.