Equivalent Expressions For Dividing Fractions A Step By Step Guide

When tackling mathematical problems, especially those involving fractions, it's crucial to understand the underlying concepts and how different operations interact. In this comprehensive guide, we'll dissect the given expression, explore the rules of dividing fractions, and meticulously evaluate each option to pinpoint the equivalent expression. Our primary goal is to not only identify the correct answer but also to illuminate the reasoning behind it, ensuring a solid grasp of the mathematical principles at play. This will involve converting mixed numbers to improper fractions, applying the division rule (multiplying by the reciprocal), and carefully handling negative signs to arrive at the accurate solution. By breaking down the problem step by step, we aim to provide a clear and concise explanation that empowers you to confidently solve similar problems in the future.

Decoding the Original Expression

The given expression is $\frac{-2 \frac{1}{4}}{-\frac{2}{3}}$. The first step in simplifying this expression is to convert the mixed number $-2 \frac{1}{4}$ into an improper fraction. To do this, we multiply the whole number part (-2) by the denominator (4) and add the numerator (1), keeping the same denominator. Thus, $-2 \frac{1}{4}$ becomes $-\frac{(2 \times 4) + 1}{4} = -\frac{8 + 1}{4} = -\frac{9}{4}$.

Now our expression looks like this: $\frac{-\frac{9}{4}}{-\frac{2}{3}}$. This is a fraction divided by a fraction, and both fractions are negative. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $-\frac{2}{3}$ is $-\frac{3}{2}$. Therefore, the expression can be rewritten as $-\frac{9}{4} \div (-\frac{2}{3}) = -\frac{9}{4} \times (-\frac{3}{2})$.

When multiplying fractions, we multiply the numerators together and the denominators together. In this case, we have $(-\frac{9}{4}) \times (-\frac{3}{2}) = \frac{(-9) \times (-3)}{4 \times 2} = \frac{27}{8}$. Note that a negative times a negative results in a positive. This simplifies our original expression to a single fraction, making it easier to compare with the answer choices provided.

To truly master the concept, let's break down the critical steps involved in handling such expressions. Firstly, recognizing and converting mixed numbers into improper fractions is paramount, as it streamlines the subsequent calculations. Secondly, understanding the division of fractions as multiplication by the reciprocal is a cornerstone principle. Lastly, meticulously managing the signs, particularly the interplay of negative signs, is crucial to prevent errors and arrive at the correct solution. By internalizing these key elements, you can confidently navigate similar problems with accuracy and efficiency, solidifying your grasp of fractional arithmetic.

Evaluating the Options

Now that we have simplified the original expression and understand its components, we will evaluate each of the provided options to determine which one is equivalent. The goal is to identify the option that, when simplified, yields the same result as our original expression, which we found to be essentially a division problem involving negative fractions. This involves understanding the division of fractions, the role of negative signs, and how these elements combine to affect the final outcome. By systematically examining each option, we can pinpoint the equivalent expression and reinforce our understanding of the underlying mathematical principles.

Option A: $\frac{9}{4} \div \frac{3}{2}$

Option A is $\frac{9}{4} \div \frac{3}{2}$. To divide fractions, we multiply by the reciprocal of the second fraction. The reciprocal of $\frac{3}{2}$ is $\frac{2}{3}$. So, the expression becomes $\frac{9}{4} \times \frac{2}{3}$. Multiplying the numerators gives us $9 \times 2 = 18$, and multiplying the denominators gives us $4 \times 3 = 12$. The result is $\frac{18}{12}$, which simplifies to $\frac{3}{2}$. This option does not seem to match our original expression's structure of negative fractions being divided, nor does it yield the same numerical result. Therefore, we can confidently eliminate option A from consideration, as it deviates both in form and value from the original problem.

Option B: $-\frac{9}{4} \div(-\frac{2}{3})$

Option B is $-\frac{9}{4} \div(-\frac{2}{3})$. As we established earlier, dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $-\frac{2}{3}$ is $-\frac{3}{2}$. Thus, the expression can be rewritten as $-\frac{9}{4} \times(-\frac{3}{2})$. Multiplying the numerators gives us $(-9) \times (-3) = 27$, and multiplying the denominators gives us $4 \times 2 = 8$. The result is $\frac{27}{8}$. This result matches the simplified form of our original expression, which we derived by converting the mixed number to an improper fraction and applying the rules of fraction division. Additionally, the presence of two negative fractions being divided aligns perfectly with the original expression's structure. Therefore, option B emerges as a strong contender for the equivalent expression.

Option C: $-\frac{9}{4} \div \frac{2}{3}$

Option C is $-\frac{9}{4} \div \frac{2}{3}$. To divide, we multiply by the reciprocal. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$. So the expression becomes $-\frac{9}{4} \times \frac{3}{2}$. Multiplying the numerators gives us $-9 \times 3 = -27$, and multiplying the denominators gives us $4 \times 2 = 8$. The result is $-\frac{27}{8}$. This result is negative, whereas our original expression simplified to a positive value. Therefore, option C is not equivalent to the original expression, as the difference in sign alone disqualifies it. We can confidently eliminate this option based on the fundamental principle that equivalent expressions must yield the same value, including the correct sign.

Option D: $\frac{9}{4} \div(-\frac{3}{2})$

Option D is $\frac{9}{4} \div(-\frac{3}{2})$. Again, dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $-\frac{3}{2}$ is $-\frac{2}{3}$. Thus, the expression becomes $\frac{9}{4} \times(-\frac{2}{3})$. Multiplying the numerators gives us $9 \times (-2) = -18$, and multiplying the denominators gives us $4 \times 3 = 12$. The result is $-\frac{18}{12}$, which simplifies to $-\frac{3}{2}$. Like option C, this result is negative, while our original expression simplified to a positive value. Therefore, option D is also not equivalent to the original expression due to the discrepancy in sign. We can eliminate this option based on the same reasoning as option C: equivalent expressions must yield the same sign, and the negative result here contradicts the positive result of the original expression.

The Verdict

After a thorough examination of each option, we've identified the expression equivalent to $\frac{-2 \frac{1}{4}}{-\frac{2}{3}}$. By systematically simplifying the original expression and comparing it to each option, we've reinforced the critical concepts of fraction division, mixed number conversion, and the handling of negative signs. This process not only leads us to the correct answer but also deepens our understanding of the underlying mathematical principles.

The correct answer is Option B: $-\frac{9}{4} \div(-\frac{2}{3})$.

This option mirrors the structure of the original expression and, when simplified, yields the same numerical value. It underscores the importance of converting mixed numbers to improper fractions, applying the principle of dividing fractions by multiplying by the reciprocal, and carefully managing the signs. This methodical approach ensures accuracy and builds confidence in tackling similar problems.

Which of the following expressions is equivalent to the fraction $\frac{-2 \frac{1}{4}}{-\frac{2}{3}}$?

Equivalent Expressions for Dividing Fractions A Step by Step Guide