Simplify The Expression: $ \frac{\frac{3}{4}+\frac{1}{8}}{\frac{5}{8}+\frac{1}{4}} $

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Introduction

In mathematics, complex fractions are a type of expression that involves multiple levels of fractions. These expressions can be challenging to simplify, but with the right approach, they can be broken down into simpler terms. In this article, we will explore how to simplify the expression: $ \frac{\frac{3}{4}+\frac{1}{8}}{\frac{5}{8}+\frac{1}{4}} $. We will use a step-by-step approach to evaluate this complex fraction and provide a clear understanding of the process.

Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In the given expression, we have a fraction in the numerator and a fraction in the denominator. To simplify this expression, we need to follow a specific order of operations.

Order of Operations

When working with complex fractions, it's essential to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Simplifying the Expression

To simplify the given expression, we will follow the order of operations:

Step 1: Evaluate the Expressions Inside Parentheses

The given expression is: $ \frac{\frac{3}{4}+\frac{1}{8}}{\frac{5}{8}+\frac{1}{4}} $

We can rewrite the expression as: $ \frac{\frac{3}{4}+\frac{1}{8}}{\frac{5}{8}+\frac{1}{4}} = \frac{\frac{6}{8}+\frac{1}{8}}{\frac{5}{8}+\frac{2}{8}} $

Now, we can simplify the fractions inside the parentheses:

$ \frac{6}{8}+\frac{1}{8} = \frac{7}{8} $

$ \frac{5}{8}+\frac{2}{8} = \frac{7}{8} $

So, the expression becomes: $ \frac{\frac{7}{8}}{\frac{7}{8}} $

Step 2: Simplify the Fraction

Now that we have simplified the expressions inside the parentheses, we can simplify the fraction:

$ \frac{\frac{7}{8}}{\frac{7}{8}} = 1 $

Therefore, the simplified expression is: $ 1 $

Conclusion

Simplifying complex fractions requires a step-by-step approach and a clear understanding of the order of operations. By following the order of operations and simplifying the expressions inside the parentheses, we can break down complex fractions into simpler terms. In this article, we have explored how to simplify the expression: $ \frac{\frac{3}{4}+\frac{1}{8}}{\frac{5}{8}+\frac{1}{4}} $ and arrived at the simplified expression: $ 1 $. We hope that this article has provided a clear understanding of the process and has been helpful in simplifying complex fractions.

Additional Tips and Resources

  • To simplify complex fractions, it's essential to follow the order of operations (PEMDAS).
  • When working with fractions, it's helpful to rewrite the fractions with a common denominator.
  • To simplify a fraction, we can multiply the numerator and denominator by the same value.
  • For more information on simplifying complex fractions, check out the following resources:
  • Khan Academy: Simplifying Complex Fractions
  • Mathway: Simplifying Complex Fractions
  • Wolfram Alpha: Simplifying Complex Fractions

Frequently Asked Questions

  • Q: What is a complex fraction? A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.
  • Q: How do I simplify a complex fraction? A: To simplify a complex fraction, follow the order of operations (PEMDAS) and simplify the expressions inside the parentheses.
  • Q: What is the order of operations (PEMDAS)? A: The order of operations (PEMDAS) is a set of rules that dictate the order in which we evaluate mathematical expressions. The acronym PEMDAS stands for:
  • P: Parentheses
  • E: Exponents
  • M: Multiplication
  • D: Division
  • A: Addition
  • S: Subtraction

Glossary of Terms

  • Complex fraction: A fraction that contains one or more fractions in its numerator or denominator.
  • Order of operations: A set of rules that dictate the order in which we evaluate mathematical expressions.
  • PEMDAS: An acronym that stands for the order of operations (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction).
  • Simplify: To reduce a complex expression to its simplest form.
    Simplify the Expression: A Q&A Guide to Complex Fractions ===========================================================

Introduction

In our previous article, we explored how to simplify the expression: $ \frac{\frac{3}{4}+\frac{1}{8}}{\frac{5}{8}+\frac{1}{4}} $. We used a step-by-step approach to evaluate this complex fraction and arrived at the simplified expression: $ 1 $. In this article, we will provide a Q&A guide to complex fractions, covering common questions and topics related to simplifying complex fractions.

Q&A Guide

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, follow the order of operations (PEMDAS) and simplify the expressions inside the parentheses.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which we evaluate mathematical expressions. The acronym PEMDAS stands for:

  • P: Parentheses
  • E: Exponents
  • M: Multiplication
  • D: Division
  • A: Addition
  • S: Subtraction

Q: How do I evaluate expressions inside parentheses?

A: To evaluate expressions inside parentheses, follow the order of operations (PEMDAS) and simplify the expressions inside the parentheses.

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

A: A simple fraction is a fraction that contains only one fraction in its numerator or denominator, while a complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Q: Can I simplify a complex fraction by multiplying the numerator and denominator by the same value?

A: Yes, you can simplify a complex fraction by multiplying the numerator and denominator by the same value. However, this method may not always result in the simplest form of the fraction.

Q: How do I know when to use the order of operations (PEMDAS)?

A: You should use the order of operations (PEMDAS) whenever you are working with complex expressions, including complex fractions.

Q: Can I simplify a complex fraction by using a calculator?

A: Yes, you can simplify a complex fraction by using a calculator. However, it's essential to understand the underlying math and the order of operations (PEMDAS) to ensure that you are getting the correct answer.

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 (PEMDAS)
  • Not simplifying expressions inside parentheses
  • Not using a common denominator when adding or subtracting fractions
  • Not multiplying the numerator and denominator by the same value when simplifying a fraction

Additional Tips and Resources

  • To simplify complex fractions, it's essential to follow the order of operations (PEMDAS).
  • When working with fractions, it's helpful to rewrite the fractions with a common denominator.
  • To simplify a fraction, we can multiply the numerator and denominator by the same value.
  • For more information on simplifying complex fractions, check out the following resources:
  • Khan Academy: Simplifying Complex Fractions
  • Mathway: Simplifying Complex Fractions
  • Wolfram Alpha: Simplifying Complex Fractions

Frequently Asked Questions

  • Q: What is a complex fraction? A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.
  • Q: How do I simplify a complex fraction? A: To simplify a complex fraction, follow the order of operations (PEMDAS) and simplify the expressions inside the parentheses.
  • Q: What is the order of operations (PEMDAS)? A: The order of operations (PEMDAS) is a set of rules that dictate the order in which we evaluate mathematical expressions. The acronym PEMDAS stands for:
  • P: Parentheses
  • E: Exponents
  • M: Multiplication
  • D: Division
  • A: Addition
  • S: Subtraction

Glossary of Terms

  • Complex fraction: A fraction that contains one or more fractions in its numerator or denominator.
  • Order of operations: A set of rules that dictate the order in which we evaluate mathematical expressions.
  • PEMDAS: An acronym that stands for the order of operations (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction).
  • Simplify: To reduce a complex expression to its simplest form.

Conclusion

Simplifying complex fractions requires a clear understanding of the order of operations (PEMDAS) and the ability to evaluate expressions inside parentheses. By following the order of operations and simplifying the expressions inside the parentheses, we can break down complex fractions into simpler terms. In this article, we have provided a Q&A guide to complex fractions, covering common questions and topics related to simplifying complex fractions. We hope that this article has been helpful in simplifying complex fractions and has provided a clear understanding of the process.