Simplify Completely: ${ \frac{\frac{1}{4}+\frac{1}{5}}{\frac{1}{4}+\frac{1}{15}} }$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 fraction. In this article, we will guide you through the process of simplifying the given expression: $\frac{\frac{1}{4}+\frac{1}{5}}{\frac{1}{4}+\frac{1}{15}}$. We will break down the problem into manageable steps and provide a clear explanation of each step.

Understanding the Problem

The given expression involves two fractions within a fraction. To simplify this expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the expressions inside the parentheses.
2. Add or subtract the fractions.
3. Divide the fractions.

Step 1: Evaluate the Expressions Inside the Parentheses

The first step is to evaluate the expressions inside the parentheses. We have two fractions inside the parentheses: $\frac{1}{4}+\frac{1}{5}$ and $\frac{1}{4}+\frac{1}{15}$.

To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 4 and 5 is 20, and the LCM of 4 and 15 is 60.

Finding a Common Denominator

To find a common denominator, we can multiply the denominators of the fractions. For the first fraction, we multiply 4 and 5 to get 20. For the second fraction, we multiply 4 and 15 to get 60.

Adding the Fractions

Now that we have a common denominator, we can add the fractions:

$\frac{1}{4}+\frac{1}{5} = \frac{5}{20}+\frac{4}{20} = \frac{9}{20}$

$\frac{1}{4}+\frac{1}{15} = \frac{15}{60}+\frac{4}{60} = \frac{19}{60}$

Step 2: Add the Fractions

Now that we have simplified the expressions inside the parentheses, we can add the fractions:

$\frac{\frac{9}{20}}{\frac{19}{60}}$

Step 3: Divide the Fractions

To divide the fractions, we can multiply the first fraction by the reciprocal of the second fraction:

$\frac{\frac{9}{20}}{\frac{19}{60}} = \frac{9}{20} \times \frac{60}{19}$

Step 4: Simplify the Expression

Now that we have multiplied the fractions, we can simplify the expression:

$\frac{9}{20} \times \frac{60}{19} = \frac{9 \times 60}{20 \times 19} = \frac{540}{380}$

Step 5: Reduce the Fraction

To reduce the fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 540 and 380 is 10.

Reducing the Fraction

To reduce the fraction, we can divide both the numerator and the denominator by the GCD:

$\frac{540}{380} = \frac{540 \div 10}{380 \div 10} = \frac{54}{38}$

Conclusion

In conclusion, the simplified expression is $\frac{54}{38}$. We can further reduce this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

Final Answer

The final answer is $\boxed{\frac{27}{19}}$.

Frequently Asked Questions

Q: What is the order of operations?

A: The order of operations is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both numbers by the GCD.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common.

Final Thoughts

Simplifying fractions can be a challenging task, but with practice and patience, you can master it. Remember to follow the order of operations and to find the greatest common divisor (GCD) of the numerator and the denominator to reduce the fraction. With these tips and the step-by-step guide provided in this article, you will be able to simplify even the most complex fractions.

Introduction

In our previous article, we provided a step-by-step guide to simplifying the expression: $\frac{\frac{1}{4}+\frac{1}{5}}{\frac{1}{4}+\frac{1}{15}}$. We also answered some frequently asked questions about simplifying fractions. In this article, we will provide a comprehensive Q&A guide to help you understand the concept of simplifying fractions.

Q&A Guide

Q: What is the order of operations?

A: The order of operations is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to follow these steps:

1. Evaluate the expressions inside the parentheses.
2. Add or subtract the fractions.
3. Divide the fractions.

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

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder.

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

A: To find the GCD of two numbers, you can use the following methods:

1. List the factors of each number and find the greatest common factor.
2. Use the Euclidean algorithm to find the GCD.
3. Use a calculator or online tool to find the GCD.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common.

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

A: To find the LCM of two numbers, you can use the following methods:

1. List the multiples of each number and find the smallest common multiple.
2. Use the formula: LCM(a, b) = (a × b) / GCD(a, b)
3. Use a calculator or online tool to find the LCM.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to follow these steps:

1. Evaluate the expressions inside the parentheses.
2. Add or subtract the fractions.
3. Divide the fractions.
4. Simplify the resulting fraction by finding the GCD of the numerator and the denominator.

Q: What is the difference between simplifying and reducing a fraction?

A: Simplifying a fraction means expressing it in its simplest form, while reducing a fraction means dividing both the numerator and the denominator by their greatest common divisor.

Q: How do I reduce a fraction?

A: To reduce a fraction, you need to find the GCD of the numerator and the denominator and divide both numbers by the GCD.

Q: What is the final answer to the expression: $\frac{\frac{1}{4}+\frac{1}{5}}{\frac{1}{4}+\frac{1}{15}}$?

A: The final answer to the expression is $\boxed{\frac{27}{19}}$.

Tips and Tricks

Tip 1: Use the order of operations to simplify complex fractions.

When simplifying complex fractions, make sure to follow the order of operations (PEMDAS) to avoid confusion.

Tip 2: Find the GCD of the numerator and the denominator to reduce the fraction.

To reduce a fraction, find the GCD of the numerator and the denominator and divide both numbers by the GCD.

Tip 3: Use a calculator or online tool to find the GCD and LCM.

If you are having trouble finding the GCD or LCM of two numbers, use a calculator or online tool to find the answer.

Conclusion

Simplifying fractions can be a challenging task, but with practice and patience, you can master it. Remember to follow the order of operations and to find the greatest common divisor (GCD) of the numerator and the denominator to reduce the fraction. With these tips and the Q&A guide provided in this article, you will be able to simplify even the most complex fractions.

Final Thoughts

Simplifying fractions is an essential skill in mathematics, and with practice, you can become proficient in it. Remember to always follow the order of operations and to find the GCD of the numerator and the denominator to reduce the fraction. With these tips and the Q&A guide provided in this article, you will be able to simplify even the most complex fractions.

Frequently Asked Questions

Q: What is the difference between simplifying and reducing a fraction?

A: Simplifying a fraction means expressing it in its simplest form, while reducing a fraction means dividing both the numerator and the denominator by their greatest common divisor.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to follow these steps:

1. Evaluate the expressions inside the parentheses.
2. Add or subtract the fractions.
3. Divide the fractions.
4. Simplify the resulting fraction by finding the GCD of the numerator and the denominator.

Q: What is the final answer to the expression: $\frac{\frac{1}{4}+\frac{1}{5}}{\frac{1}{4}+\frac{1}{15}}$?

A: The final answer to the expression is $\boxed{\frac{27}{19}}$.

Additional Resources

Online Tools

• GCD calculator: www.gcdcalculator.com
• LCM calculator: www.lcmcalculator.com
• Fraction simplifier: www.fractionsimplifier.com

Books

• "Algebra and Trigonometry" by Michael Sullivan
• "Mathematics for Dummies" by Mary Jane Sterling
• "Calculus for Dummies" by Mark Ryan

Websites

• Khan Academy: www.khanacademy.org
• Mathway: www.mathway.com
• Wolfram Alpha: www.wolframalpha.com