Simplify The Expression:$\[ -\frac{38}{27} \div \frac{57}{18} \\]

Feb 26, 2025 by ADMIN 66 views

Simplify the Expression: A Step-by-Step Guide to Dividing Fractions

Introduction

When it comes to simplifying expressions involving fractions, it's essential to understand the rules and procedures for dividing fractions. In this article, we'll delve into the world of fraction division and provide a step-by-step guide on how to simplify the expression: $-\frac{38}{27} \div \frac{57}{18}$ .

Understanding Fraction Division

Before we dive into the solution, let's take a moment to understand the concept of fraction division. When we divide one fraction by another, we're essentially asking how many times the first fraction fits into the second fraction. This can be a bit tricky, but with the right approach, it becomes manageable.

The Rule for Dividing Fractions

The rule for dividing fractions is straightforward: to divide one fraction by another, we multiply the first fraction by the reciprocal of the second fraction. In other words, if we have $\frac{a}{b} \div \frac{c}{d}$ , we can rewrite it as $\frac{a}{b} \times \frac{d}{c}$ .

Applying the Rule to the Given Expression

Now that we've covered the basics, let's apply the rule to the given expression: $-\frac{38}{27} \div \frac{57}{18}$ . To simplify this expression, we'll multiply the first fraction by the reciprocal of the second fraction.

Step 1: Multiply the First Fraction by the Reciprocal of the Second Fraction

To simplify the expression, we'll multiply $-\frac{38}{27}$ by the reciprocal of $\frac{57}{18}$ , which is $\frac{18}{57}$ .

Step 2: Multiply the Numerators and Denominators

Now that we have the fractions set up, we can multiply the numerators and denominators. The numerator of the first fraction is $-38$ , and the denominator is $27$ . The numerator of the second fraction is $18$ , and the denominator is $57$ .

Step 3: Simplify the Resulting Fraction

After multiplying the numerators and denominators, we get: $\frac{-38 \times 18}{27 \times 57}$ . Now, we can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor (GCD).

Step 4: Find the Greatest Common Divisor (GCD)

To simplify the fraction, we need to find the GCD of $-684$ and $1539$ . The GCD is the largest number that divides both numbers without leaving a remainder.

Step 5: Simplify the Fraction

After finding the GCD, we can simplify the fraction by dividing both the numerator and denominator by the GCD. In this case, the GCD is $3$ , so we can divide both $-684$ and $1539$ by $3$ .

Step 6: Write the Final Answer

After simplifying the fraction, we get: $\frac{-228}{513}$ .

Conclusion

In this article, we've covered the basics of fraction division and provided a step-by-step guide on how to simplify the expression: $-\frac{38}{27} \div \frac{57}{18}$ . By following the rule for dividing fractions and simplifying the resulting fraction, we arrived at the final answer: $\frac{-228}{513}$ . We hope this article has provided valuable insights and helped you understand the concept of fraction division.

Frequently Asked Questions

What is the rule for dividing fractions?
How do I simplify a fraction after dividing?
What is the greatest common divisor (GCD), and how do I find it?

Additional Resources

Final Thoughts

Simplifying expressions involving fractions can be a challenging task, but with the right approach and practice, it becomes manageable. By following the rule for dividing fractions and simplifying the resulting fraction, you can arrive at the final answer with confidence. Remember to always find the greatest common divisor (GCD) and simplify the fraction to its simplest form. With this knowledge, you'll be well on your way to becoming a master of fraction division.

Introduction

In our previous article, we covered the basics of fraction division and provided a step-by-step guide on how to simplify the expression: $-\frac{38}{27} \div \frac{57}{18}$ . However, we know that there are many more questions and concerns that you may have. In this article, we'll address some of the most frequently asked questions about simplifying expressions involving fractions.

Q&A: Simplifying Expressions Involving Fractions

Q: What is the rule for dividing fractions?

A: The rule for dividing fractions is straightforward: to divide one fraction by another, we multiply the first fraction by the reciprocal of the second fraction. In other words, if we have $\frac{a}{b} \div \frac{c}{d}$ , we can rewrite it as $\frac{a}{b} \times \frac{d}{c}$ .

Q: How do I simplify a fraction after dividing?

A: To simplify a fraction after dividing, we need to find the greatest common divisor (GCD) of the numerator and denominator. We can then divide both the numerator and denominator by the GCD to simplify the fraction.

Q: What is the greatest common divisor (GCD), and how do I find it?

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and denominator of a fraction without leaving a remainder. To find the GCD, we can use the Euclidean algorithm or list the factors of the numerator and denominator and find the greatest common factor.

Q: How do I handle negative fractions when dividing?

A: When dividing negative fractions, we need to remember that a negative divided by a negative is a positive. So, if we have $-\frac{a}{b} \div -\frac{c}{d}$ , we can rewrite it as $\frac{a}{b} \div \frac{c}{d}$ .

Q: Can I simplify a fraction before dividing?

A: Yes, we can simplify a fraction before dividing. In fact, it's often easier to simplify the fraction before dividing. We can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD).

Q: What if the GCD of the numerator and denominator is 1?

A: If the GCD of the numerator and denominator is 1, then the fraction is already in its simplest form. In this case, we don't need to simplify the fraction further.

Q: Can I use a calculator to simplify fractions?

A: Yes, we can use a calculator to simplify fractions. However, it's always a good idea to check the answer by hand to make sure it's correct.

Conclusion

In this article, we've addressed some of the most frequently asked questions about simplifying expressions involving fractions. We hope this article has provided valuable insights and helped you understand the concept of fraction division. Remember to always follow the rule for dividing fractions and simplify the resulting fraction to its simplest form.