Mastering Fraction Operations Solving B. 8 ÷ (-14/5) × (-7) And E. 18 ÷ (-9/2) × (-3/4)

In the realm of mathematics, a solid understanding of fundamental operations is crucial for tackling more complex problems. This article delves into the intricacies of dividing and multiplying fractions, providing a comprehensive guide to help you master these essential skills. We will explore the concepts, techniques, and real-world applications of these operations, equipping you with the knowledge and confidence to excel in your mathematical endeavors. Specifically, we will dissect two example problems b. 8 ÷ ${\left(-\frac{14}{5}\right) × (-7)}$ and e. 18 ÷ {\left(-\frac{9}{2}\right) × \(\left(-\frac{3}{4}\right)}, offering step-by-step solutions and explanations to solidify your understanding.

Understanding Division and Multiplication of Fractions

Before diving into the specific examples, let's establish a firm grasp of the underlying principles of dividing and multiplying fractions. A fraction represents a part of a whole, expressed as a ratio of two numbers: the numerator (top number) and the denominator (bottom number). Multiplication of fractions involves multiplying the numerators and the denominators separately. For example, to multiply ${\frac{a}{b}}$ by ${\frac{c}{d}}$, we calculate ${\frac{a × c}{b × d}}$. Division, on the other hand, is the inverse operation of multiplication. Dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of a fraction ${\frac{a}{b}}$ is ${\frac{b}{a}}$. Therefore, to divide ${\frac{a}{b}}$ by ${\frac{c}{d}}$, we multiply ${\frac{a}{b}}$ by ${\frac{d}{c}}$, resulting in ${\frac{a × d}{b × c}}$. These rules form the bedrock of fraction arithmetic and are essential for solving a wide range of mathematical problems. The order of operations, often remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction), dictates that multiplication and division are performed from left to right. This principle is particularly important when dealing with expressions involving both operations, as it ensures that the calculations are carried out in the correct sequence.

Problem b: 8 ÷ (-14/5) × (-7)

Let's tackle the first problem: 8 ÷ ${\left(-\frac{14}{5}\right) × (-7)}$. This problem combines division and multiplication of fractions, requiring us to apply the principles discussed earlier. The first step is to convert the whole number 8 into a fraction by expressing it as ${\frac{8}{1}}$. Next, we address the division operation. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of ${\left(-\frac{14}{5}\right)}$ is ${\left(-\frac{5}{14}\right)}$. Therefore, we rewrite the expression as ${\frac{8}{1} × \left(-\frac{5}{14}\right) × (-7)}$. Now, we can perform the multiplication. Multiplying ${\frac{8}{1}}$ by ${\left(-\frac{5}{14}\right)}$ gives us ${\left(-\frac{40}{14}\right)}$. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This simplifies the fraction to ${\left(-\frac{20}{7}\right)}$. Finally, we multiply ${\left(-\frac{20}{7}\right)}$ by (-7). To do this, we can express -7 as the fraction ${\left(-\frac{7}{1}\right)}$. Multiplying the two fractions, we get ${\frac{140}{7}}$. Simplifying this fraction by dividing both numerator and denominator by 7, we arrive at the final answer: 20. Therefore, 8 ÷ ${\left(-\frac{14}{5}\right) × (-7) = 20}$. Understanding the rules of multiplying and dividing signed numbers is crucial here. A negative number multiplied by a negative number yields a positive number, which explains why the final result is positive 20.

Problem e: 18 ÷ (-9/2) × (-3/4)

Now, let's dissect the second problem: 18 ÷ {\left(-\frac{9}{2}\right) × \(\left(-\frac{3}{4}\right)}. Similar to the previous example, this problem involves both division and multiplication of fractions. We begin by converting the whole number 18 into a fraction, writing it as ${\frac{18}{1}}$. The next step is to address the division operation. We divide by a fraction by multiplying by its reciprocal. The reciprocal of ${\left(-\frac{9}{2}\right)}$ is ${\left(-\frac{2}{9}\right)}$. Thus, the expression becomes ${\frac{18}{1} × \left(-\frac{2}{9}\right) × \left(-\frac{3}{4}\right)}$. We now proceed with the multiplication. Multiplying ${\frac{18}{1}}$ by ${\left(-\frac{2}{9}\right)}$ gives us ${\left(-\frac{36}{9}\right)}$. This fraction can be simplified by dividing both the numerator and the denominator by 9, resulting in -4. We can express -4 as the fraction ${\left(-\frac{4}{1}\right)}$. Finally, we multiply ${\left(-\frac{4}{1}\right)}$ by ${\left(-\frac{3}{4}\right)}$. Multiplying these two fractions yields ${\frac{12}{4}}$. Simplifying this fraction by dividing both the numerator and the denominator by 4, we obtain the final answer: 3. Consequently, 18 ÷ {\left(-\frac{9}{2}\right) × \(\left(-\frac{3}{4}\right) = 3}. This example further reinforces the importance of understanding the rules of signs in multiplication and division. The product of two negative numbers is positive, which is why the final answer is positive 3.

Step-by-Step Solutions Explained

To further clarify the process, let's revisit the solutions to both problems with a more granular, step-by-step explanation:

Problem b: 8 ÷ (-14/5) × (-7)

1. Convert the whole number to a fraction: 8 becomes ${\frac{8}{1}}$.
2. Rewrite division as multiplication by the reciprocal: The reciprocal of ${\left(-\frac{14}{5}\right)}$ is ${\left(-\frac{5}{14}\right)}$. The expression becomes ${\frac{8}{1} × \left(-\frac{5}{14}\right) × (-7)}$.
3. Multiply the first two fractions: ${\frac{8}{1} × \left(-\frac{5}{14}\right) = \left(-\frac{40}{14}\right)}$.
4. Simplify the fraction: ${\left(-\frac{40}{14}\right)}$ simplifies to ${\left(-\frac{20}{7}\right)}$.
5. Convert the whole number to a fraction: -7 becomes ${\left(-\frac{7}{1}\right)}$.
6. Multiply the resulting fraction by the last term: ${\left(-\frac{20}{7}\right) × \left(-\frac{7}{1}\right) = \frac{140}{7}}$.
7. Simplify the fraction: ${\frac{140}{7}}$ simplifies to 20.

Therefore, 8 ÷ ${\left(-\frac{14}{5}\right) × (-7) = 20}$.

Problem e: 18 ÷ (-9/2) × (-3/4)

1. Convert the whole number to a fraction: 18 becomes ${\frac{18}{1}}$.
2. Rewrite division as multiplication by the reciprocal: The reciprocal of ${\left(-\frac{9}{2}\right)}$ is ${\left(-\frac{2}{9}\right)}$. The expression becomes ${\frac{18}{1} × \left(-\frac{2}{9}\right) × \left(-\frac{3}{4}\right)}$.
3. Multiply the first two fractions: ${\frac{18}{1} × \left(-\frac{2}{9}\right) = \left(-\frac{36}{9}\right)}$.
4. Simplify the fraction: ${\left(-\frac{36}{9}\right)}$ simplifies to -4, which can be written as ${\left(-\frac{4}{1}\right)}$.
5. Multiply the resulting fraction by the last term: ${\left(-\frac{4}{1}\right) × \left(-\frac{3}{4}\right) = \frac{12}{4}}$.
6. Simplify the fraction: ${\frac{12}{4}}$ simplifies to 3.

Therefore, 18 ÷ {\left(-\frac{9}{2}\right) × \(\left(-\frac{3}{4}\right) = 3}.

Common Mistakes and How to Avoid Them

When working with division and multiplication of fractions, several common mistakes can arise. Recognizing these pitfalls and understanding how to avoid them is crucial for achieving accuracy and building confidence in your mathematical abilities. One frequent error is forgetting to invert the second fraction when dividing. Remember, dividing by a fraction is equivalent to multiplying by its reciprocal. Failing to perform this inversion will lead to an incorrect result. Another common mistake is incorrectly applying the rules of signs. When multiplying or dividing numbers with different signs, the result is negative. When the signs are the same, the result is positive. It's essential to keep these rules in mind to avoid sign errors. A third mistake stems from not simplifying fractions before or after performing operations. Simplifying fractions makes the calculations easier and prevents working with unnecessarily large numbers. Always look for opportunities to simplify fractions by dividing both the numerator and the denominator by their greatest common divisor. Finally, errors can occur due to misapplication of the order of operations. Remember PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Ensure you perform operations in the correct sequence to arrive at the accurate answer.

Real-World Applications

The concepts of dividing and multiplying fractions are not confined to the classroom; they have numerous practical applications in everyday life. Consider scenarios involving cooking and baking, where recipes often call for scaling ingredients up or down. This requires multiplying or dividing fractional quantities to maintain the correct proportions. In construction and carpentry, measurements frequently involve fractions, and accurately calculating lengths and areas often necessitates multiplying and dividing fractions. Financial calculations, such as determining portions of investments or calculating interest rates, also rely on fraction arithmetic. Moreover, in fields like science and engineering, many formulas and equations involve fractions, making a solid understanding of these operations essential for problem-solving and analysis. For instance, calculating the concentration of a solution in chemistry or determining the efficiency of a machine in engineering often involves working with fractions. The ability to confidently and accurately manipulate fractions is therefore a valuable skill that extends far beyond the realm of theoretical mathematics.

Conclusion

Mastering the division and multiplication of fractions is a fundamental step towards achieving mathematical proficiency. By understanding the underlying principles, practicing diligently, and avoiding common mistakes, you can develop the skills necessary to tackle a wide range of mathematical problems. The step-by-step solutions to the example problems presented in this article, 8 ÷ ${\left(-\frac{14}{5}\right) × (-7)}$ and 18 ÷ {\left(-\frac{9}{2}\right) × \(\left(-\frac{3}{4}\right)}, provide a solid foundation for further exploration. Remember to always simplify fractions, pay attention to the order of operations, and apply the rules of signs correctly. With consistent effort and a clear understanding of the concepts, you can confidently navigate the world of fractions and unlock their power in both academic and real-world applications.