Simplify The Expression: $\frac{\frac{2+x}{8x}}{\frac{3+x}{x}}$Choose The Correct Simplified Form:A. $\frac{2+x}{8(3+x)}$B. $\frac{2+x}{8}$C. $\frac{2+x}{3+x}$D. $\frac{2+x}{8x^2}$

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Understanding the Problem

When simplifying complex expressions, it's essential to break them down into manageable parts. In this case, we're given the expression $\frac{\frac{2+x}{8x}}{\frac{3+x}{x}}$. Our goal is to simplify this expression to its most basic form.

Step 1: Simplify the Complex Fraction

To simplify the complex fraction, we need to divide the numerator by the denominator. This can be achieved by multiplying the numerator by the reciprocal of the denominator.

$\frac{\frac{2+x}{8x}}{\frac{3+x}{x}} = \frac{2+x}{8x} \cdot \frac{x}{3+x}$

Step 2: Multiply the Numerators and Denominators

Now, we multiply the numerators and denominators separately.

$\frac{(2+x) \cdot x}{8x \cdot (3+x)}$

Step 3: Simplify the Expression

We can simplify the expression by canceling out common factors in the numerator and denominator.

$\frac{(2+x) \cdot x}{8x \cdot (3+x)} = \frac{(2+x)}{8(3+x)}$

Step 4: Choose the Correct Simplified Form

Now that we have simplified the expression, we can compare it to the given options.

• Option A: $\frac{2+x}{8(3+x)}$
• Option B: $\frac{2+x}{8}$
• Option C: $\frac{2+x}{3+x}$
• Option D: $\frac{2+x}{8x^2}$

Based on our simplified expression, the correct answer is:

A. $\frac{2+x}{8(3+x)}$

Conclusion

Simplifying complex expressions requires breaking them down into manageable parts and applying the rules of arithmetic operations. By following the steps outlined above, we can simplify the expression $\frac{\frac{2+x}{8x}}{\frac{3+x}{x}}$ to its most basic form.

Tips and Tricks

• When simplifying complex fractions, it's essential to multiply the numerator by the reciprocal of the denominator.
• Cancel out common factors in the numerator and denominator to simplify the expression.
• Compare the simplified expression to the given options to choose the correct answer.

Common Mistakes

• Failing to multiply the numerator by the reciprocal of the denominator.
• Not canceling out common factors in the numerator and denominator.
• Choosing the wrong option based on the simplified expression.

Real-World Applications

Simplifying complex expressions is a crucial skill in various fields, including mathematics, physics, and engineering. It's essential to be able to simplify expressions to their most basic form to solve problems and make informed decisions.

Practice Problems

• Simplify the expression: $\frac{\frac{3x}{4y}}{\frac{2x}{y}}$
• Simplify the expression: $\frac{\frac{5x}{6y}}{\frac{3x}{2y}}$

Solutions

• $\frac{3x}{4y} \cdot \frac{y}{2x} = \frac{3}{8}$
• $\frac{5x}{6y} \cdot \frac{2y}{3x} = \frac{5}{9}$
  

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Frequently Asked Questions

Q: What is the correct simplified form of the expression $\frac{\frac{2+x}{8x}}{\frac{3+x}{x}}$?

A: The correct simplified form of the expression is $\frac{2+x}{8(3+x)}$.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to divide the numerator by the denominator. This can be achieved by multiplying the numerator by the reciprocal of the 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 $\frac{a}{b}$ is $\frac{b}{a}$.

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

A: To multiply the numerator by the reciprocal of the denominator, you need to multiply the numerator by the denominator of the reciprocal fraction. For example, to multiply $\frac{a}{b}$ by $\frac{c}{d}$, you need to multiply $a$ by $d$ and $b$ by $c$.

Q: What is the difference between multiplying the numerator by the reciprocal of the denominator and dividing the numerator by the denominator?

A: Multiplying the numerator by the reciprocal of the denominator is equivalent to dividing the numerator by the denominator. This is because the reciprocal of a fraction is obtained by swapping the numerator and denominator.

Q: How do I simplify the expression $\frac{\frac{3x}{4y}}{\frac{2x}{y}}$?

A: To simplify the expression, you need to multiply the numerator by the reciprocal of the denominator. This gives you $\frac{3x}{4y} \cdot \frac{y}{2x} = \frac{3}{8}$.

Q: How do I simplify the expression $\frac{\frac{5x}{6y}}{\frac{3x}{2y}}$?

A: To simplify the expression, you need to multiply the numerator by the reciprocal of the denominator. This gives you $\frac{5x}{6y} \cdot \frac{2y}{3x} = \frac{5}{9}$.

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

A: Some common mistakes to avoid when simplifying complex fractions include failing to multiply the numerator by the reciprocal of the denominator, not canceling out common factors in the numerator and denominator, and choosing the wrong option based on the simplified expression.

Q: What are some real-world applications of simplifying complex fractions?

A: Simplifying complex fractions is a crucial skill in various fields, including mathematics, physics, and engineering. It's essential to be able to simplify expressions to their most basic form to solve problems and make informed decisions.

Additional Resources

• Khan Academy: Simplifying Complex Fractions
• Mathway: Simplifying Complex Fractions
• Wolfram Alpha: Simplifying Complex Fractions

Practice Problems

• Simplify the expression: $\frac{\frac{7x}{9y}}{\frac{3x}{y}}$
• Simplify the expression: $\frac{\frac{11x}{12y}}{\frac{4x}{2y}}$

Solutions

• $\frac{7x}{9y} \cdot \frac{y}{3x} = \frac{7}{27}$
• $\frac{11x}{12y} \cdot \frac{2y}{4x} = \frac{11}{24}$