Simplify The Expression: $\[\frac{4}{9} \div 2\\]

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Introduction

Understanding Division of Fractions

When dealing with division of fractions, it's essential to remember that dividing by a fraction is the same as multiplying by its reciprocal. In this case, we have the expression ${\frac{4}{9} \div 2}$. To simplify this expression, we need to apply the rules of division and fractions.

The Rules of Division and Fractions

Reciprocal Rule

The reciprocal rule states that dividing by a fraction is the same as multiplying by its reciprocal. In other words, ${\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}}$. This rule will be the foundation of our simplification process.

Applying the Reciprocal Rule

To simplify the expression $\frac{4}{9} \div 2}$, we need to apply the reciprocal rule. Since we are dividing by a whole number (2), we can rewrite it as a fraction with a denominator of 1 ${2 = \frac{2{1}}$. Now, we can apply the reciprocal rule:

49รท2=49ร—12{\frac{4}{9} \div 2 = \frac{4}{9} \times \frac{1}{2}}

Multiplying Fractions

Multiplying Numerators and Denominators

When multiplying fractions, we multiply the numerators together and the denominators together. In this case, we have:

49ร—12=4ร—19ร—2{\frac{4}{9} \times \frac{1}{2} = \frac{4 \times 1}{9 \times 2}}

Simplifying the Expression

Simplifying the Numerator and Denominator

Now that we have multiplied the fractions, we can simplify the expression by dividing the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 4 and 18 is 2. Dividing both the numerator and denominator by 2, we get:

4ร—19ร—2=29{\frac{4 \times 1}{9 \times 2} = \frac{2}{9}}

Conclusion

In conclusion, we have simplified the expression ${\frac{4}{9} \div 2}$ using the reciprocal rule and the rules of multiplication and division of fractions. The final simplified expression is ${\frac{2}{9}}$. This process demonstrates the importance of understanding the rules of division and fractions in simplifying complex expressions.

Additional Examples

Simplifying More Complex Expressions

To further illustrate the concept of simplifying expressions using the reciprocal rule, let's consider a more complex example:

34รท56{\frac{3}{4} \div \frac{5}{6}}

Using the reciprocal rule, we can rewrite this expression as:

34ร—65{\frac{3}{4} \times \frac{6}{5}}

Multiplying the fractions, we get:

3ร—64ร—5=1820{\frac{3 \times 6}{4 \times 5} = \frac{18}{20}}

Simplifying the expression by dividing the numerator and denominator by their GCD (2), we get:

1820=910{\frac{18}{20} = \frac{9}{10}}

This example demonstrates how the reciprocal rule can be applied to more complex expressions involving division of fractions.

Final Thoughts

The Importance of Understanding Division of Fractions

In conclusion, understanding the rules of division and fractions is crucial in simplifying complex expressions. The reciprocal rule provides a foundation for simplifying expressions involving division of fractions. By applying this rule and the rules of multiplication and division of fractions, we can simplify expressions and arrive at a final answer. This process requires a deep understanding of the underlying mathematical concepts and principles.

Frequently Asked Questions

Common Questions and Answers

Q: What is the reciprocal rule? A: The reciprocal rule states that dividing by a fraction is the same as multiplying by its reciprocal.

Q: How do I apply the reciprocal rule? A: To apply the reciprocal rule, rewrite the divisor as a fraction with a denominator of 1, and then multiply the fractions.

Q: What is the greatest common divisor (GCD)? A: The GCD is the largest number that divides both the numerator and denominator of a fraction.

Q: How do I simplify a fraction? A: To simplify a fraction, divide the numerator and denominator by their GCD.

References

  • [1] "Division of Fractions" by Math Open Reference
  • [2] "Reciprocal Rule" by Khan Academy
  • [3] "Greatest Common Divisor" by Wolfram MathWorld

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources.

Introduction

Understanding Division of Fractions

When dealing with division of fractions, it's essential to remember that dividing by a fraction is the same as multiplying by its reciprocal. In this article, we'll answer some of the most common questions related to simplifying expressions with division of fractions.

Q&A

Q: What is the reciprocal rule?

A: The Reciprocal Rule Explained

The reciprocal rule states that dividing by a fraction is the same as multiplying by its reciprocal. In other words, ${\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}}$. This rule is the foundation of simplifying expressions involving division of fractions.

Q: How do I apply the reciprocal rule?

A: Applying the Reciprocal Rule

To apply the reciprocal rule, rewrite the divisor as a fraction with a denominator of 1, and then multiply the fractions. For example, to simplify the expression ${\frac{4}{9} \div 2}$, we can rewrite 2 as ${2 = \frac{2}{1}}$. Then, we can multiply the fractions:

49รท2=49ร—12{\frac{4}{9} \div 2 = \frac{4}{9} \times \frac{1}{2}}

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

A: Understanding the Greatest Common Divisor

The GCD is the largest number that divides both the numerator and denominator of a fraction. For example, the GCD of 4 and 18 is 2. To simplify a fraction, we can divide the numerator and denominator by their GCD.

Q: How do I simplify a fraction?

A: Simplifying Fractions

To simplify a fraction, divide the numerator and denominator by their GCD. For example, to simplify the fraction ${\frac{18}{20}}$, we can divide both the numerator and denominator by 2:

1820=910{\frac{18}{20} = \frac{9}{10}}

Q: What is the difference between dividing by a fraction and multiplying by its reciprocal?

A: Dividing by a Fraction vs. Multiplying by its Reciprocal

Dividing by a fraction is the same as multiplying by its reciprocal. In other words, ${\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}}$. This rule is the foundation of simplifying expressions involving division of fractions.

Q: Can I simplify an expression with multiple fractions?

A: Simplifying Expressions with Multiple Fractions

Yes, you can simplify an expression with multiple fractions by applying the reciprocal rule and the rules of multiplication and division of fractions. For example, to simplify the expression ${\frac{3}{4} \div \frac{5}{6}}$, we can apply the reciprocal rule:

34รท56=34ร—65{\frac{3}{4} \div \frac{5}{6} = \frac{3}{4} \times \frac{6}{5}}

Then, we can multiply the fractions and simplify the expression:

3ร—64ร—5=1820{\frac{3 \times 6}{4 \times 5} = \frac{18}{20}}

Simplifying the expression by dividing the numerator and denominator by their GCD (2), we get:

1820=910{\frac{18}{20} = \frac{9}{10}}

Conclusion

In conclusion, understanding the rules of division and fractions is crucial in simplifying complex expressions. The reciprocal rule provides a foundation for simplifying expressions involving division of fractions. By applying this rule and the rules of multiplication and division of fractions, we can simplify expressions and arrive at a final answer. This process requires a deep understanding of the underlying mathematical concepts and principles.

Additional Resources

  • [1] "Division of Fractions" by Math Open Reference
  • [2] "Reciprocal Rule" by Khan Academy
  • [3] "Greatest Common Divisor" by Wolfram MathWorld

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources.