Simplify Completely:${ \frac{\frac{1}{2}+\frac{1}{7}}{\frac{1}{2}+\frac{1}{14}} }$Write Your Answer As An Integer Or A Reduced Fraction In The Form { \frac{A}{B}$}$.

Introduction

When dealing with fractions, simplifying them can be a daunting task, especially when they involve multiple fractions within a single expression. In this article, we will focus on simplifying the given expression: $\frac{\frac{1}{2}+\frac{1}{7}}{\frac{1}{2}+\frac{1}{14}}$. We will break down the process into manageable steps, using a combination of mathematical techniques and logical reasoning to arrive at the final answer.

Understanding the Expression

Before we dive into simplifying the expression, let's take a closer look at what we're dealing with. The given expression is a complex fraction, which means it has a fraction in both the numerator and the denominator. Our goal is to simplify this expression by reducing it to its simplest form, which is an integer or a reduced fraction in the form $\frac{A}{B}$.

Step 1: Simplify the Numerator

To simplify the expression, we need to start by simplifying the numerator. The numerator is the fraction $\frac{1}{2}+\frac{1}{7}$. To add these two fractions, we need to find a common denominator, which is the least common multiple (LCM) of 2 and 7. The LCM of 2 and 7 is 14.

# Simplify the Numerator
Step 1: Find the Common Denominator
The common denominator of 2 and 7 is 14.
Step 2: Rewrite the Fractions with the Common Denominator
$\frac{1}{2} = \frac{7}{14}$ and $\frac{1}{7} = \frac{2}{14}$
Step 3: Add the Fractions
$\frac{7}{14} + \frac{2}{14} = \frac{9}{14}$

Step 2: Simplify the Denominator

Now that we have simplified the numerator, let's move on to simplifying the denominator. The denominator is the fraction $\frac{1}{2}+\frac{1}{14}$. Again, we need to find a common denominator, which is the LCM of 2 and 14. The LCM of 2 and 14 is 14.

# Simplify the Denominator
Step 1: Find the Common Denominator
The common denominator of 2 and 14 is 14.
Step 2: Rewrite the Fractions with the Common Denominator
$\frac{1}{2} = \frac{7}{14}$ and $\frac{1}{14} = \frac{1}{14}$
Step 3: Add the Fractions
$\frac{7}{14} + \frac{1}{14} = \frac{8}{14}$

Step 3: Simplify the Expression

Now that we have simplified the numerator and the denominator, we can simplify the expression by dividing the numerator by the denominator. To do this, we need to invert the denominator and multiply.

# Simplify the Expression
Step 1: Invert the Denominator
$\frac{8}{14} = \frac{2}{7}$
Step 2: Multiply the Numerator by the Inverted Denominator
$\frac{9}{14} \times \frac{7}{2} = \frac{63}{28}$

Step 4: Reduce the Fraction

The final step is to reduce the fraction to its simplest form. To do this, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 63 and 28 is 7.

# Reduce the Fraction
Step 1: Find the Greatest Common Divisor (GCD)
The GCD of 63 and 28 is 7.
Step 2: Divide the Numerator and the Denominator by the GCD
$\frac{63}{28} = \frac{9}{4}$

Conclusion

In conclusion, we have successfully simplified the given expression: $\frac{\frac{1}{2}+\frac{1}{7}}{\frac{1}{2}+\frac{1}{14}}$. By breaking down the process into manageable steps, we were able to arrive at the final answer, which is a reduced fraction in the form $\frac{A}{B}$. The final answer is $\boxed{\frac{9}{4}}$.

Final Answer

The final answer is $\boxed{\frac{9}{4}}$.

Introduction

In our previous article, we explored the process of simplifying a complex fraction by reducing it to its simplest form. We walked through the steps of simplifying the numerator and the denominator, and finally arrived at the final answer. In this article, we will take a Q&A approach to further clarify the concepts and provide additional insights into simplifying fractions.

Q&A: Simplifying Fractions

Q: What is the first step in simplifying a complex fraction?

A: The first step in simplifying a complex fraction is to simplify the numerator. This involves finding a common denominator for the fractions in the numerator and adding them together.

Q: How do I find the common denominator for two fractions?

A: To find the common denominator for two fractions, you need to find the least common multiple (LCM) of the two denominators. The LCM is the smallest number that both denominators can divide into evenly.

Q: What is the difference between a common denominator and a least common multiple?

A: A common denominator is a number that both fractions can divide into evenly, while a least common multiple is the smallest number that both fractions can divide into evenly.

Q: How do I simplify the denominator of a complex fraction?

A: To simplify the denominator of a complex fraction, you need to find a common denominator for the fractions in the denominator and add them together.

Q: What is the final step in simplifying a complex fraction?

A: The final step in simplifying a complex fraction is to reduce the fraction to its simplest form. This involves dividing the numerator and the denominator by their greatest common divisor (GCD).

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that both the numerator and the denominator can divide into evenly.

Q: How do I find the GCD of two numbers?

A: To find the GCD of two numbers, you can use the Euclidean algorithm or simply list the factors of each number and find the largest common factor.

Q: What is the final answer to the complex fraction $\frac{\frac{1}{2}+\frac{1}{7}}{\frac{1}{2}+\frac{1}{14}}$?

A: The final answer to the complex fraction $\frac{\frac{1}{2}+\frac{1}{7}}{\frac{1}{2}+\frac{1}{14}}$ is $\boxed{\frac{9}{4}}$.

Additional Tips and Tricks

Tip 1: Simplify the numerator and the denominator separately

When simplifying a complex fraction, it's often easier to simplify the numerator and the denominator separately before combining them.

Tip 2: Use a common denominator to add fractions

When adding fractions, it's essential to use a common denominator to ensure that the fractions are added correctly.

Tip 3: Reduce the fraction to its simplest form

After simplifying the numerator and the denominator, be sure to reduce the fraction to its simplest form by dividing the numerator and the denominator by their GCD.

Conclusion

In conclusion, simplifying complex fractions can be a challenging task, but by breaking it down into manageable steps and using the right techniques, you can arrive at the final answer. Remember to simplify the numerator and the denominator separately, use a common denominator to add fractions, and reduce the fraction to its simplest form. With practice and patience, you'll become a pro at simplifying complex fractions in no time!

Final Answer

The final answer is $\boxed{\frac{9}{4}}$.