Mathias Found The Product \[$\frac{4x+8}{x} \cdot \frac{5}{x+2}\$\] As Follows:$\[ \frac{4x+8}{x} \cdot \frac{5}{x+2} = \frac{4x+8}{x} \cdot \frac{5}{x+2} = \frac{12}{1} \cdot \frac{5}{x+2} = \frac{60}{x+2} \\]Which Of The Following

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Introduction

In mathematics, simplifying complex expressions is a crucial skill that helps us solve problems efficiently. Mathias, a math enthusiast, has come across a product of two fractions and has attempted to simplify it. In this article, we will analyze Mathias' work and provide a step-by-step guide on how to simplify the given product.

The Given Product

The product Mathias is working with is:

4x+8x⋅5x+2\frac{4x+8}{x} \cdot \frac{5}{x+2}

Mathias' Solution

Mathias has attempted to simplify the product as follows:

4x+8x⋅5x+2=4x+8x⋅5x+2=121⋅5x+2=60x+2\frac{4x+8}{x} \cdot \frac{5}{x+2} = \frac{4x+8}{x} \cdot \frac{5}{x+2} = \frac{12}{1} \cdot \frac{5}{x+2} = \frac{60}{x+2}

Analysis of Mathias' Solution

At first glance, Mathias' solution seems correct. However, let's break it down step by step to ensure that it is indeed correct.

Step 1: Factor out the GCF

The first step in simplifying the product is to factor out the greatest common factor (GCF) from the numerator and denominator of each fraction. In this case, the GCF of 4x+84x+8 is 44, and the GCF of xx is 11. Similarly, the GCF of 55 is 55, and the GCF of x+2x+2 is 11.

import sympy as sp

# Define the variables
x = sp.symbols('x')

# Define the fractions
f1 = (4*x + 8) / x
f2 = 5 / (x + 2)

# Factor out the GCF
f1_factored = sp.factor(f1)
f2_factored = sp.factor(f2)

print(f1_factored)
print(f2_factored)

Step 2: Multiply the Fractions

Once we have factored out the GCF, we can multiply the fractions together.

# Multiply the fractions
product = f1_factored * f2_factored

print(product)

Step 3: Simplify the Product

After multiplying the fractions, we can simplify the product by canceling out any common factors.

# Simplify the product
simplified_product = sp.simplify(product)

print(simplified_product)

Conclusion

In conclusion, Mathias' solution is not entirely correct. While he has attempted to simplify the product, he has made a mistake in the process. By following the steps outlined above, we can see that the correct simplification of the product is:

4x+8x⋅5x+2=20x⋅5x+2=100x(x+2)\frac{4x+8}{x} \cdot \frac{5}{x+2} = \frac{20}{x} \cdot \frac{5}{x+2} = \frac{100}{x(x+2)}

Therefore, the final answer is 100x(x+2)\boxed{\frac{100}{x(x+2)}}.

Discussion

This problem highlights the importance of carefully following the steps in simplifying complex expressions. Mathias' mistake was in not factoring out the GCF correctly and not canceling out the common factors. By following the steps outlined above, we can ensure that we simplify expressions correctly and avoid making mistakes.

Additional Resources

For more practice problems on simplifying expressions, check out the following resources:

  • Khan Academy: Simplifying Expressions
  • Mathway: Simplifying Expressions
  • Wolfram Alpha: Simplifying Expressions

Final Thoughts

Q&A: Simplifying Products of Fractions

In the previous article, we analyzed Mathias' attempt to simplify a product of two fractions. In this article, we will provide a Q&A section to help you better understand the concept of simplifying products of fractions.

Q: What is the first step in simplifying a product of fractions?

A: The first step in simplifying a product of fractions is to factor out the greatest common factor (GCF) from the numerator and denominator of each fraction.

Q: How do I factor out the GCF from a fraction?

A: To factor out the GCF from a fraction, you need to identify the GCF of the numerator and denominator. Then, you can rewrite the fraction with the GCF factored out.

Q: What is the next step in simplifying a product of fractions?

A: After factoring out the GCF, the next step is to multiply the fractions together.

Q: How do I multiply fractions together?

A: To multiply fractions together, you need to multiply the numerators and denominators separately. Then, you can simplify the resulting fraction by canceling out any common factors.

Q: What is the final step in simplifying a product of fractions?

A: The final step in simplifying a product of fractions is to simplify the resulting fraction by canceling out any common factors.

Q: Can you provide an example of simplifying a product of fractions?

A: Let's consider the product:

4x+8x⋅5x+2\frac{4x+8}{x} \cdot \frac{5}{x+2}

To simplify this product, we need to factor out the GCF from each fraction:

4x+8x=4(x+2)x\frac{4x+8}{x} = \frac{4(x+2)}{x}

5x+2=5x+2\frac{5}{x+2} = \frac{5}{x+2}

Then, we can multiply the fractions together:

4(x+2)x⋅5x+2=20x\frac{4(x+2)}{x} \cdot \frac{5}{x+2} = \frac{20}{x}

Finally, we can simplify the resulting fraction by canceling out any common factors:

20x=20x\frac{20}{x} = \frac{20}{x}

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

A: Some common mistakes to avoid when simplifying products of fractions include:

  • Not factoring out the GCF correctly
  • Not canceling out common factors
  • Not simplifying the resulting fraction

Q: How can I practice simplifying products of fractions?

A: You can practice simplifying products of fractions by working through examples and exercises. You can also use online resources, such as Khan Academy or Mathway, to practice simplifying products of fractions.

Conclusion

In conclusion, simplifying products of fractions is a crucial skill in mathematics that helps us solve problems efficiently. By following the steps outlined above, we can ensure that we simplify expressions correctly and avoid making mistakes. Remember to always carefully follow the steps and check your work to ensure that you are getting the correct answer.

Additional Resources

For more practice problems on simplifying products of fractions, check out the following resources:

  • Khan Academy: Simplifying Products of Fractions
  • Mathway: Simplifying Products of Fractions
  • Wolfram Alpha: Simplifying Products of Fractions

Final Thoughts

Simplifying products of fractions is a fundamental concept in mathematics that helps us solve problems efficiently. By following the steps outlined above and practicing regularly, you can become proficient in simplifying products of fractions and tackle more complex problems with confidence.