Find The Quotient. Simplify Your Answer.${ \frac{n+5}{n} \div \frac{n 2-25}{8n 2} }$

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Understanding Complex Fractions

Complex fractions are a type of mathematical expression that involves fractions within fractions. They can be challenging to simplify, but with the right approach, you can break them down and find the quotient. In this article, we will explore the process of simplifying complex fractions, using the given expression as an example.

The Given Expression

The given expression is n+5n÷n2−258n2\frac{n+5}{n} \div \frac{n^2-25}{8n^2}. To simplify this expression, we need to follow the order of operations (PEMDAS) and apply the rules of division and fractions.

Step 1: Simplify the Dividend and Divisor

The first step is to simplify the dividend and divisor separately. The dividend is n+5n\frac{n+5}{n}, and the divisor is n2−258n2\frac{n^2-25}{8n^2}.

Simplifying the Dividend

To simplify the dividend, we can factor the numerator and denominator. The numerator can be factored as n+5n+5, and the denominator can be factored as nn. Therefore, the simplified dividend is n+5n\frac{n+5}{n}.

Simplifying the Divisor

To simplify the divisor, we can factor the numerator and denominator. The numerator can be factored as (n+5)(n−5)(n+5)(n-5), and the denominator can be factored as 8n28n^2. Therefore, the simplified divisor is (n+5)(n−5)8n2\frac{(n+5)(n-5)}{8n^2}.

Step 2: Apply the Quotient Rule

The quotient rule states that when dividing two fractions, we can multiply the first fraction by the reciprocal of the second fraction. In this case, we can multiply the dividend by the reciprocal of the divisor.

Multiplying the Dividend by the Reciprocal of the Divisor

The reciprocal of the divisor is 8n2(n+5)(n−5)\frac{8n^2}{(n+5)(n-5)}. Therefore, we can multiply the dividend by the reciprocal of the divisor:

n+5n⋅8n2(n+5)(n−5)\frac{n+5}{n} \cdot \frac{8n^2}{(n+5)(n-5)}

Step 3: Simplify the Expression

Now that we have multiplied the dividend by the reciprocal of the divisor, we can simplify the expression. We can cancel out common factors in the numerator and denominator.

Canceling Out Common Factors

The numerator and denominator have a common factor of (n+5)(n+5). We can cancel out this factor:

n+5n⋅8n2(n+5)(n−5)=n+5n⋅8n2(n−5)\frac{n+5}{n} \cdot \frac{8n^2}{(n+5)(n-5)} = \frac{n+5}{n} \cdot \frac{8n^2}{(n-5)}

Step 4: Simplify the Expression Further

Now that we have canceled out the common factor, we can simplify the expression further. We can cancel out the common factor of nn in the numerator and denominator:

n+5n⋅8n2(n−5)=n+51⋅8n(n−5)\frac{n+5}{n} \cdot \frac{8n^2}{(n-5)} = \frac{n+5}{1} \cdot \frac{8n}{(n-5)}

Step 5: Simplify the Expression to Find the Quotient

Now that we have simplified the expression, we can find the quotient. We can multiply the numerator and denominator:

n+51⋅8n(n−5)=(n+5)(8n)(n−5)\frac{n+5}{1} \cdot \frac{8n}{(n-5)} = \frac{(n+5)(8n)}{(n-5)}

The Final Answer

The final answer is 8n2+40nn−5\boxed{\frac{8n^2+40n}{n-5}}.

Conclusion

Simplifying complex fractions requires a step-by-step approach. By following the order of operations and applying the rules of division and fractions, we can break down complex fractions and find the quotient. In this article, we used the given expression as an example and simplified it to find the quotient. With practice and patience, you can master the art of simplifying complex fractions and tackle even the most challenging mathematical expressions.

Common Mistakes to Avoid

When simplifying complex fractions, there are several common mistakes to avoid:

  • Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying complex fractions.
  • Not canceling out common factors: Make sure to cancel out common factors in the numerator and denominator.
  • Not simplifying the expression further: Make sure to simplify the expression further by canceling out common factors and multiplying the numerator and denominator.

Tips and Tricks

When simplifying complex fractions, here are some tips and tricks to keep in mind:

  • Use the quotient rule: The quotient rule states that when dividing two fractions, we can multiply the first fraction by the reciprocal of the second fraction.
  • Cancel out common factors: Make sure to cancel out common factors in the numerator and denominator.
  • Simplify the expression further: Make sure to simplify the expression further by canceling out common factors and multiplying the numerator and denominator.

Real-World Applications

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

  • Finance: Simplifying complex fractions is essential in finance, where complex financial expressions need to be simplified to make informed decisions.
  • Engineering: Simplifying complex fractions is essential in engineering, where complex mathematical expressions need to be simplified to design and build complex systems.
  • Science: Simplifying complex fractions is essential in science, where complex mathematical expressions need to be simplified to understand and describe complex phenomena.

Conclusion

Simplifying complex fractions is a crucial skill in mathematics and has many real-world applications. By following the order of operations and applying the rules of division and fractions, we can break down complex fractions and find the quotient. With practice and patience, you can master the art of simplifying complex fractions and tackle even the most challenging mathematical expressions.

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 understand the process of simplifying complex fractions.

Q: What is a complex fraction?

A: A complex fraction is a type of mathematical expression that involves fractions within fractions. It is a fraction that contains one or more fractions in the numerator or denominator.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to follow the order of operations (PEMDAS) and apply the rules of division and fractions. You can start by simplifying the dividend and divisor separately, and then apply the quotient rule to simplify the expression.

Q: What is the quotient rule?

A: The quotient rule states that when dividing two fractions, you can multiply the first fraction by the reciprocal of the second fraction. This rule is essential in simplifying complex fractions.

Q: How do I cancel out common factors in a complex fraction?

A: To cancel out common factors in a complex fraction, you need to identify the common factors in the numerator and denominator and cancel them out. This will help you simplify the expression and find the quotient.

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

A: Some common mistakes to avoid when simplifying complex fractions include not following the order of operations, not canceling out common factors, and not simplifying the expression further. Make sure to follow the order of operations and cancel out common factors to simplify the expression.

Q: How do I know if a complex fraction is simplified?

A: A complex fraction is simplified when there are no common factors left in the numerator and denominator. You can check if a complex fraction is simplified by multiplying the numerator and denominator and seeing if the result is a whole number or a simple fraction.

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

A: While a calculator can be helpful in simplifying complex fractions, it is not always necessary. You can use a calculator to check your work and ensure that the expression is simplified correctly. However, it is essential to understand the process of simplifying complex fractions and be able to do it manually.

Q: How do I apply the quotient rule to simplify a complex fraction?

A: To apply the quotient rule, you need to multiply the first fraction by the reciprocal of the second fraction. This will help you simplify the expression and find the quotient.

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

A: Simplifying complex fractions has many real-world applications, including finance, engineering, and science. In finance, complex fractions are used to calculate interest rates and investment returns. In engineering, complex fractions are used to design and build complex systems. In science, complex fractions are used to understand and describe complex phenomena.

Q: Can I simplify complex fractions with variables?

A: Yes, you can simplify complex fractions with variables. However, you need to follow the same rules and procedures as you would with numerical fractions. You can use the quotient rule and cancel out common factors to simplify the expression.

Q: How do I check my work when simplifying complex fractions?

A: To check your work, you can multiply the numerator and denominator and see if the result is a whole number or a simple fraction. You can also use a calculator to check your work and ensure that the expression is simplified correctly.

Q: What are some tips and tricks for simplifying complex fractions?

A: Some tips and tricks for simplifying complex fractions include using the quotient rule, canceling out common factors, and simplifying the expression further. You can also use a calculator to check your work and ensure that the expression is simplified correctly.

Conclusion

Simplifying complex fractions is a crucial skill in mathematics and has many real-world applications. By following the order of operations and applying the rules of division and fractions, you can break down complex fractions and find the quotient. With practice and patience, you can master the art of simplifying complex fractions and tackle even the most challenging mathematical expressions.