Evaluate The Expression:$\[ \frac{1}{2} \div \frac{2}{3} \times \frac{1}{4} = \\]

Introduction

In mathematics, expressions involving fractions can be complex and challenging to simplify. One common operation that can be performed on fractions is division, which can be denoted by the symbol ÷. In this article, we will evaluate the expression $\frac{1}{2} \div \frac{2}{3} \times \frac{1}{4}$ and provide a step-by-step guide on how to simplify it.

Understanding the Order of Operations

Before we begin evaluating the expression, it's essential to understand the order of operations, which is a set of rules that dictate the order in which mathematical operations should be performed. The order of operations is often remembered using the acronym PEMDAS, which stands for:

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.

Evaluating the Expression

Now that we have a basic understanding of the order of operations, let's begin evaluating the expression $\frac{1}{2} \div \frac{2}{3} \times \frac{1}{4}$.

Step 1: Invert and Multiply

To divide two fractions, we need to invert the second fraction and multiply it by the first fraction. In this case, we will invert $\frac{2}{3}$ and multiply it by $\frac{1}{2}$.

$\frac{1}{2} \div \frac{2}{3} = \frac{1}{2} \times \frac{3}{2}$

Step 2: Multiply the Numerators and Denominators

Now that we have the expression $\frac{1}{2} \times \frac{3}{2}$, we can multiply the numerators and denominators.

$\frac{1}{2} \times \frac{3}{2} = \frac{1 \times 3}{2 \times 2}$

Step 3: Simplify the Expression

Now that we have the expression $\frac{3}{4}$, we can simplify it by dividing the numerator and denominator by their greatest common divisor (GCD).

$\frac{3}{4} = \frac{3 \div 1}{4 \div 1}$

Step 4: Multiply by the Remaining Fraction

Now that we have the expression $\frac{3}{4}$, we can multiply it by the remaining fraction $\frac{1}{4}$.

$\frac{3}{4} \times \frac{1}{4} = \frac{3 \times 1}{4 \times 4}$

Step 5: Simplify the Final Expression

Now that we have the expression $\frac{3}{16}$, we can simplify it by dividing the numerator and denominator by their GCD.

$\frac{3}{16} = \frac{3 \div 1}{16 \div 1}$

Conclusion

In this article, we evaluated the expression $\frac{1}{2} \div \frac{2}{3} \times \frac{1}{4}$ and provided a step-by-step guide on how to simplify it. We used the order of operations to evaluate the expression, inverting and multiplying the fractions, multiplying the numerators and denominators, simplifying the expression, and multiplying by the remaining fraction. The final expression was $\frac{3}{16}$.

Frequently Asked Questions

• Q: What is the order of operations? A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The order of operations is often remembered using the acronym PEMDAS.
• Q: How do I divide two fractions? A: To divide two fractions, you need to invert the second fraction and multiply it by the first fraction.
• Q: How do I simplify a fraction? A: To simplify a fraction, you need to divide the numerator and denominator by their greatest common divisor (GCD).

Additional Resources

• Khan Academy: Order of Operations
• Mathway: Simplifying Fractions
• Wolfram Alpha: Evaluating Expressions

Final Thoughts

Evaluating complex expressions involving fractions can be challenging, but with the right tools and techniques, it can be done. By following the order of operations and using the techniques outlined in this article, you can simplify even the most complex expressions. Remember to invert and multiply fractions, multiply the numerators and denominators, simplify the expression, and multiply by the remaining fraction. With practice and patience, you will become proficient in evaluating complex expressions and simplifying fractions.

Introduction

Evaluating expressions and simplifying fractions can be a challenging task, especially for those who are new to mathematics. In this article, we will answer some of the most frequently asked questions related to evaluating expressions and simplifying fractions.

Q&A

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The order of operations is often remembered using the acronym PEMDAS, which stands for:

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.

Q: How do I divide two fractions?

A: To divide two fractions, you need to invert the second fraction and multiply it by the first fraction. For example, to divide $\frac{1}{2}$ by $\frac{2}{3}$, you would invert $\frac{2}{3}$ to get $\frac{3}{2}$ and then multiply it by $\frac{1}{2}$.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to divide the numerator and denominator by their greatest common divisor (GCD). For example, to simplify $\frac{6}{8}$, you would divide both the numerator and denominator by 2 to get $\frac{3}{4}$.

Q: What is the difference between a numerator and a denominator?

A: The numerator is the top number in a fraction, while the denominator is the bottom number. For example, in the fraction $\frac{3}{4}$, 3 is the numerator and 4 is the denominator.

Q: How do I multiply fractions?

A: To multiply fractions, you need to multiply the numerators and denominators separately. For example, to multiply $\frac{1}{2}$ and $\frac{3}{4}$, you would multiply the numerators to get 3 and the denominators to get 8, resulting in $\frac{3}{8}$.

Q: Can I simplify a fraction by dividing both the numerator and denominator by the same number?

A: Yes, you can simplify a fraction by dividing both the numerator and denominator by the same number. For example, to simplify $\frac{6}{8}$, you can divide both the numerator and denominator by 2 to get $\frac{3}{4}$.

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

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and denominator of a fraction without leaving a remainder. For example, the GCD of 6 and 8 is 2.

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

A: There are several ways to find the GCD of two numbers, including:

• Listing the factors of each number and finding the greatest common factor
• Using the Euclidean algorithm
• Using a calculator or online tool

Q: Can I simplify a fraction by multiplying both the numerator and denominator by the same number?

A: Yes, you can simplify a fraction by multiplying both the numerator and denominator by the same number. For example, to simplify $\frac{1}{2}$, you can multiply both the numerator and denominator by 2 to get $\frac{2}{4}$, which can then be simplified to $\frac{1}{2}$.

Conclusion

Evaluating expressions and simplifying fractions can be a challenging task, but with the right tools and techniques, it can be done. By following the order of operations and using the techniques outlined in this article, you can simplify even the most complex expressions. Remember to invert and multiply fractions, multiply the numerators and denominators, simplify the expression, and multiply by the remaining fraction. With practice and patience, you will become proficient in evaluating complex expressions and simplifying fractions.

Additional Resources

• Khan Academy: Order of Operations
• Mathway: Simplifying Fractions
• Wolfram Alpha: Evaluating Expressions
• GCD Calculator: Find the greatest common divisor of two numbers

Final Thoughts

Evaluating complex expressions and simplifying fractions is an essential skill in mathematics. By mastering these skills, you will be able to solve a wide range of mathematical problems and become proficient in evaluating complex expressions. Remember to practice regularly and seek help when needed. With dedication and persistence, you will become a master of evaluating expressions and simplifying fractions.