Evaluating (1/2) ÷ (-1/3) ÷ (-2/5) A Step By Step Guide

In the realm of mathematics, evaluating expressions involving fractions is a fundamental skill. This article will provide a comprehensive, step-by-step guide on how to evaluate the expression ${ \frac{1}{2} \div \left(-\frac{1}{3}\right) \div \frac{-2}{5} }$, ensuring a clear understanding of the underlying concepts and techniques. We will break down each step, explaining the logic behind it and providing helpful insights to enhance your problem-solving abilities. This is crucial for anyone studying basic arithmetic, algebra, or more advanced mathematical topics. Understanding how to manipulate fractions and perform division operations is a cornerstone of mathematical literacy, impacting your ability to solve complex equations, analyze data, and even perform everyday calculations. Therefore, mastering this skill will not only benefit you academically but also in practical situations.

When tackling expressions with multiple divisions, it’s essential to approach them methodically. This involves understanding the order of operations and how division interacts with negative numbers. The expression ${ \frac{1}{2} \div \left(-\frac{1}{3}\right) \div \frac{-2}{5} }$ presents a combination of fractions and negative signs, making it a perfect example to illustrate these concepts. We will explore how to convert division into multiplication by inverting the divisor and how to handle the negative signs to arrive at the correct answer. Moreover, this guide will also emphasize the importance of simplifying fractions whenever possible, as this can make the calculations easier and less prone to errors. By following this detailed explanation, you will gain confidence in evaluating similar expressions and develop a deeper understanding of the properties of fractions and division.

This article aims to make the process as clear and straightforward as possible, providing not just the steps but also the rationale behind them. Whether you are a student learning the basics or someone looking to refresh your mathematical skills, this guide will offer valuable insights and techniques to improve your understanding and proficiency in working with fractions and division. So, let's delve into the expression and unravel the steps to evaluate it accurately.

Step 1: Convert Division to Multiplication

The first step in evaluating the expression ${ \frac{1}{2} \div \left(-\frac{1}{3}\right) \div \frac{-2}{5} }$ is to convert the division operations into multiplication. This is a crucial step because dividing by a fraction is mathematically equivalent to multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator. For instance, the reciprocal of ${ \frac{a}{b} }$ is ${ \frac{b}{a} }$.

Applying this principle to our expression, we start with the first division: ${ \frac{1}{2} \div \left(-\frac{1}{3}\right) }$. To convert this division to multiplication, we take the reciprocal of { -rac{1}{3} }, which is { -rac{3}{1} } or simply -3. Therefore, ${ \frac{1}{2} \div \left(-\frac{1}{3}\right) }$ becomes ${ \frac{1}{2} \times (-3) }$. This transformation is fundamental in simplifying the expression and making it easier to solve. The key concept here is that division is the inverse operation of multiplication, and by using the reciprocal, we are essentially performing the inverse operation in a more convenient way.

Next, we perform the multiplication: ${ \frac{1}{2} \times (-3) = \frac{1 \times (-3)}{2 \times 1} = \frac{-3}{2} }$. This step involves multiplying the numerators together and the denominators together. In this case, 1 multiplied by -3 gives -3, and 2 multiplied by 1 gives 2. So, the result of the first division and multiplication is ${ \frac{-3}{2} }$. This result is then carried forward to the next operation in the expression. This systematic approach of converting division to multiplication and then performing the multiplication is a cornerstone of working with fractions and is essential for solving more complex expressions.

Now, we have simplified the expression to ${ \frac{-3}{2} \div \frac{-2}{5} }$. We still have a division operation to handle, so we repeat the process. We convert the division to multiplication by taking the reciprocal of ${ \frac{-2}{5} }$, which is ${ \frac{-5}{2} }$. Thus, ${ \frac{-3}{2} \div \frac{-2}{5} }$ becomes ${ \frac{-3}{2} \times \frac{-5}{2} }$. This step further simplifies the expression and sets us up for the final multiplication, which will give us the final result. By consistently applying the principle of converting division to multiplication, we can efficiently solve expressions involving multiple divisions.

Step 2: Perform the Multiplication

Having converted the division operations into multiplication, we now focus on performing the multiplication ${ \frac{-3}{2} \times \frac{-5}{2} }$. To multiply fractions, we multiply the numerators together and the denominators together. This is a straightforward process that allows us to combine the fractions into a single fraction. In this case, we have ${ (-3) \times (-5) }$ for the numerator and ${ 2 \times 2 }$ for the denominator. It’s important to pay close attention to the signs, as the product of two negative numbers is a positive number. This rule is fundamental in arithmetic and ensures that we arrive at the correct sign for the final answer.

Let's calculate the numerator: ${ (-3) \times (-5) = 15 }$. The product of two negative numbers, -3 and -5, is indeed positive 15. This is a critical point to remember when dealing with negative numbers in multiplication and division. The sign rules are essential for maintaining accuracy in mathematical calculations. Now, let's calculate the denominator: ${ 2 \times 2 = 4 }$. This is a simple multiplication, and the result is 4.

So, the result of the multiplication ${ \frac{-3}{2} \times \frac{-5}{2} }$ is ${ \frac{15}{4} }$. This fraction represents the final value of the expression. However, it's often helpful to express the result in different forms, such as a mixed number or a decimal, depending on the context or the specific requirements of the problem. We will explore this in the next step to provide a more complete understanding of the solution.

At this point, it's beneficial to review the steps we've taken so far. We started by converting the division operations to multiplication using the reciprocal. Then, we performed the multiplication by multiplying the numerators and the denominators. This systematic approach is crucial for solving more complex expressions and avoiding common errors. The ability to confidently multiply fractions and handle negative signs is a key skill in mathematics, and this step-by-step explanation should help reinforce that skill.

Step 3: Express the Result as a Mixed Number (Optional)

While ${ \frac{15}{4} }$ is a perfectly valid answer, it is an improper fraction, meaning the numerator is greater than the denominator. In many contexts, it is preferable to express improper fractions as mixed numbers. A mixed number consists of a whole number and a proper fraction (where the numerator is less than the denominator). Converting an improper fraction to a mixed number provides a different perspective on the value and can sometimes make it easier to understand or compare with other values.

To convert ${ \frac{15}{4} }$ to a mixed number, we divide the numerator (15) by the denominator (4). The quotient becomes the whole number part of the mixed number, and the remainder becomes the numerator of the fractional part, with the original denominator remaining the same. So, let's perform the division: 15 divided by 4. 4 goes into 15 three times (${ 4 \times 3 = 12 }$), with a remainder of 3 (${ 15 - 12 = 3 }$).

Therefore, the whole number part of our mixed number is 3, and the remainder 3 becomes the numerator of the fractional part, with the denominator remaining 4. This gives us the mixed number ${ 3 \frac{3}{4} }$. This means that ${ \frac{15}{4} }$ is equivalent to 3 whole units and three-quarters of another unit. Expressing the result as a mixed number can provide a more intuitive understanding of the quantity, especially in real-world applications where whole units and fractional parts are commonly encountered.

In summary, ${ \frac{15}{4} }$ can be expressed as the mixed number ${ 3 \frac{3}{4} }$. This conversion is a useful skill to have, as it allows you to present your answer in a format that is often easier to interpret. Whether you leave your answer as an improper fraction or convert it to a mixed number depends on the specific requirements of the problem or the context in which you are working. Both forms are mathematically correct, but the mixed number can sometimes provide a clearer sense of the magnitude of the value.

Step 4: Final Answer

After methodically working through each step, we have successfully evaluated the expression ${ \frac{1}{2} \div \left(-\frac{1}{3}\right) \div \frac{-2}{5} }$. By converting the division operations into multiplication and carefully performing the calculations, we have arrived at the final answer. It's essential to reiterate the importance of each step to ensure a clear understanding of the process. We began by converting division to multiplication, then performed the multiplication, and finally, we expressed the result as both an improper fraction and a mixed number.

The final result of the expression is ${ \frac{15}{4} }$, which can also be expressed as the mixed number ${ 3 \frac{3}{4} }$. This answer represents the value obtained by performing the given operations in the correct order. It is crucial to understand that the order of operations is paramount in mathematics, and any deviation from this order can lead to an incorrect result. In this case, we followed the standard convention of performing division from left to right after converting it to multiplication.

This step-by-step guide has not only provided the solution but also aimed to enhance your understanding of the underlying concepts. Evaluating expressions involving fractions and division is a fundamental skill in mathematics, and mastering this skill will significantly improve your problem-solving abilities in various mathematical contexts. From basic arithmetic to more advanced algebra and calculus, the ability to manipulate fractions and perform division operations accurately is essential.

In conclusion, the expression ${ \frac{1}{2} \div \left(-\frac{1}{3}\right) \div \frac{-2}{5} }$ evaluates to ${ \frac{15}{4} }$ or ${ 3 \frac{3}{4} }$. This result underscores the importance of following the correct order of operations and understanding the properties of fractions and division. With practice and a clear understanding of the principles involved, you can confidently tackle similar expressions and excel in your mathematical endeavors.

In summary, evaluating the expression ${ \frac{1}{2} \div \left(-\frac{1}{3}\right) \div \frac{-2}{5} }$ demonstrates the crucial steps involved in simplifying mathematical expressions with fractions and division. The process includes converting division to multiplication by using reciprocals, performing the multiplication of fractions, and expressing the result in both improper fraction and mixed number forms. Each of these steps highlights fundamental mathematical principles that are essential for problem-solving in various contexts.

The final answer, ${ \frac{15}{4} }$ or ${ 3 \frac{3}{4} }$, not only provides the numerical solution but also underscores the significance of understanding mathematical operations and their order. By methodically working through each step, we reinforce the importance of accuracy and attention to detail in mathematical calculations. This exercise serves as a valuable learning tool for students and anyone looking to refresh their mathematical skills.

Ultimately, mastering the evaluation of expressions like this builds a strong foundation for more complex mathematical concepts. The skills and techniques discussed here are applicable across a wide range of mathematical disciplines, from basic arithmetic to advanced calculus. By practicing and internalizing these principles, individuals can develop greater confidence and proficiency in mathematics, which is essential for both academic and real-world applications.

By following this guide, you have gained a comprehensive understanding of how to evaluate expressions involving fractions and division. This knowledge will empower you to tackle more complex problems and further enhance your mathematical skills. Remember, practice is key to mastery, so continue to apply these principles to various mathematical problems to solidify your understanding and build your confidence.