Equivalent Division: Solving $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$

Hey everyone, let's dive into a neat math problem! We're tasked with finding the equivalent division expression for the fraction $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$. Don't worry, it's not as scary as it looks. We'll break it down step-by-step, making it super easy to understand. This is a common type of question you'll encounter in math, especially when dealing with fractions and division. The goal is to rewrite the given expression in a different format while maintaining its original value. This helps in simplifying calculations and understanding the underlying mathematical concepts. So, let's get started and find the correct equivalent expression from the options given. The core concept here revolves around the rules of dividing fractions, and understanding how mixed numbers work. This exercise helps in solidifying your grasp on these fundamental concepts. By the end of this, you'll be a pro at identifying equivalent division expressions!

Converting the Mixed Number to an Improper Fraction

Our first step, and a super important one, is to convert the mixed number into an improper fraction. Remember, mixed numbers are those that have a whole number part and a fractional part (like $4 \frac{1}{3}$). Improper fractions, on the other hand, have a numerator larger than their denominator (like $\frac{13}{3}$). Why do we do this? Because it makes the division process much smoother. Let's convert $4 \frac{1}{3}$ to an improper fraction.

To do this, we multiply the whole number (4) by the denominator of the fraction (3), and then add the numerator (1). The result becomes the new numerator, and we keep the original denominator.

So, $(4 \times 3) + 1 = 12 + 1 = 13$. Therefore, $4 \frac{1}{3}$ becomes $\frac{13}{3}$. Great job, everyone! We've taken the first step toward simplifying our expression. This conversion is a crucial skill in fraction manipulation, making complex expressions easier to manage. Remember that understanding this process is essential for many math problems that involve fractions. Practice these conversions frequently to become more comfortable and confident. It’s like learning a secret code that unlocks easier solutions to math problems! Now, let's substitute this back into our original expression.

Rewriting the Original Expression

Now that we've converted the mixed number, let's rewrite the original expression with the improper fraction. The original expression was $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$. After converting $4 \frac{1}{3}$ to $\frac{13}{3}$, our expression becomes $\frac{\frac{13}{3}}{-\frac{5}{6}}$. This is now a fraction divided by a fraction. Remember, the negative sign can be placed in the numerator, the denominator, or in front of the entire fraction. All three positions are equivalent and do not change the value of the fraction.

So, our expression can be rewritten as $\frac{\frac{13}{3}}{-\frac{5}{6}}$, which is the same as $\frac{13}{3} \div \left(-\frac{5}{6}\right)$. We're now dealing with a simple division problem. This format makes it clear how to proceed with the division operation. Keep in mind that understanding how to rewrite expressions is important. This helps you to manipulate equations and simplifies problem-solving. It's akin to having multiple tools in your toolkit, making you versatile in tackling different math challenges. Remember, the goal is to make the problem easier to solve. Always look for ways to simplify and rewrite equations to make them more manageable.

Identifying the Correct Equivalent Expression

Now, let's look at the options provided to find the equivalent division expression. We've simplified our original expression to $\frac{13}{3} \div \left(-\frac{5}{6}\right)$. Let’s go through the answer choices to identify the correct equivalent division expression.

• Option A: $\frac{13}{3} \div \left(-\frac{5}{6}\right)$: This is exactly what we derived! The improper fraction $\frac{13}{3}$ is being divided by $-\frac{5}{6}$. Bingo! This is the correct answer. The negative sign is crucial, so always pay attention to its placement. So, this option perfectly matches our simplified expression.

• Option B: $-\frac{5}{6} \div \frac{13}{3}$: This expression has the numbers in the wrong order and the incorrect operation, this will lead to a completely different result. Remember that the order of division matters.

• Option C: $\frac{13}{3} \div \frac{5}{6}$: This expression is very close, but it’s missing the negative sign. The original expression had a negative sign in the denominator. This is not the correct equivalent division expression.

• Option D: $-\frac{13}{3} - \left(-\frac{5}{6}\right)$: This uses subtraction, not division. This is not at all equivalent to our initial expression. The operations must match for the expressions to be considered equivalent.

So, we've carefully evaluated each option. It's clear that Option A is the correct answer. Understanding how to handle negative signs and the order of operations is key. Always double-check your work to avoid making simple mistakes. Taking your time and being careful is always the best strategy. Keep practicing these types of problems, and you'll become a pro in no time! Remember to always convert mixed numbers to improper fractions, and pay close attention to the placement of negative signs. This detailed approach ensures that you select the correct answer and understand why other options are incorrect. By following these steps, you'll be well-prepared to tackle similar problems in the future.

Conclusion: Finding the Equivalent Expression

Alright, guys! We've made it to the end. The equivalent division expression for $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ is $\frac{13}{3} \div \left(-\frac{5}{6}\right)$. We successfully converted the mixed number, rewrote the expression, and then carefully examined the provided options. You've now learned how to tackle these types of fraction problems. Remember, the key is to stay organized, convert mixed numbers, and carefully consider the signs. Keep practicing, and you'll ace these problems in no time. Always double-check your work, and don't be afraid to ask for help if you need it. Math can be fun! Believe in yourselves, and keep up the amazing work! You are now equipped with the knowledge to solve these types of problems confidently.

In this example, Option A, $\frac{13}{3} \div \left(-\frac{5}{6}\right)$, is the correct equivalent division expression. The ability to manipulate and rewrite expressions is a valuable skill in mathematics. It allows for simplifying calculations and better understanding of the problem. Always remember to break down complex problems into smaller, more manageable steps. Practice regularly to improve your skills in fraction manipulation and division. You got this, keep up the great work! Always review your work to ensure accuracy and understanding. Great job, everyone!