Simplifying: (-4) * (7/5) * (-3/4) / (7/15)

In this article, we will walk through the step-by-step process of simplifying the expression $(-4) \times (\frac{7}{5}) \times (-\frac{3}{4}) \div (\frac{7}{15})$. This involves multiplication and division of integers and fractions. Understanding the order of operations and how to manipulate fractions is crucial for solving this problem efficiently. Let’s dive in!

Step 1: Understanding the Order of Operations

Before we begin, it's important to remember the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our expression, we only have multiplication and division, which should be performed from left to right. This means we'll first multiply $(-4)$ by $(\frac{7}{5})$, then multiply the result by $(-\frac{3}{4})$, and finally divide by $(\frac{7}{15})$. Getting this order correct is essential to arriving at the correct answer. When dealing with a mix of negative numbers and fractions, keeping track of the signs is also very important. A negative times a negative results in a positive, and a negative times a positive results in a negative. By following these rules and understanding the basics, you will become proficient in solving such mathematical problems, building a solid foundation for more complex calculations in the future. Keep practicing, and you'll find these operations become second nature. Remember, accuracy and attention to detail are key in mathematics. Double-check each step to avoid common mistakes such as sign errors or incorrect fraction manipulations. With persistence and focus, you can master these skills and confidently tackle more challenging problems.

Step 2: Multiplying the First Two Terms

Let's start by multiplying the first two terms: $(-4) \times (\frac{7}{5})$. To do this, we can rewrite $-4$ as a fraction: $(-4) = (\frac{-4}{1})$. Now, we multiply the numerators and the denominators:

$(\frac{-4}{1}) \times (\frac{7}{5}) = \frac{(-4) \times 7}{1 \times 5} = \frac{-28}{5}$.

So, the result of the first multiplication is $-\frac{28}{5}$. When multiplying fractions, always remember to multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator. This principle applies regardless of whether the numbers are positive or negative. In this case, we multiplied $-4$ by $7$ to get $-28$ for the numerator, and $1$ by $5$ to get $5$ for the denominator. Therefore, the product is $-\frac{28}{5}$. This foundational step is crucial because it simplifies the expression, paving the way for the subsequent operations. Accuracy in this initial step prevents errors from propagating through the rest of the calculation, ensuring a correct final answer. Pay close attention to signs and ensure you are comfortable with the multiplication of fractions before moving forward.

Step 3: Multiplying by the Third Term

Next, we multiply the result from Step 2, which is $-\frac{28}{5}$, by the third term, which is $-\frac{3}{4}$:

$(-\frac{28}{5}) \times (-\frac{3}{4}) = \frac{(-28) \times (-3)}{5 \times 4} = \frac{84}{20}$.

Since both numbers are negative, the result is positive. Now, we can simplify the fraction $\frac{84}{20}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

$\frac{84 \div 4}{20 \div 4} = \frac{21}{5}$.

So, the result of multiplying the first three terms is $\frac{21}{5}$. When multiplying two negative fractions, remember that the product is always positive. This is a fundamental rule of arithmetic that helps maintain accuracy in calculations. In this step, we multiplied $-\frac{28}{5}$ by $-\frac{3}{4}$ and obtained $\frac{84}{20}$. To simplify this fraction, we found the greatest common divisor (GCD) of 84 and 20, which is 4. Dividing both the numerator and the denominator by 4 gives us the simplified fraction $\frac{21}{5}$. Simplifying fractions makes them easier to work with in subsequent steps and also presents the final answer in its most concise form. Regularly practicing these simplification techniques will enhance your mathematical skills and increase your problem-solving efficiency.

Step 4: Dividing by the Last Term

Now, we need to divide the result from Step 3, which is $\frac{21}{5}$, by the last term, which is $\frac{7}{15}$. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{7}{15}$ is $\frac{15}{7}$. Therefore, we have:

$\frac{21}{5} \div \frac{7}{15} = \frac{21}{5} \times \frac{15}{7}$.

Multiplying these fractions:

$\frac{21}{5} \times \frac{15}{7} = \frac{21 \times 15}{5 \times 7} = \frac{315}{35}$.

Now, we simplify the fraction $\frac{315}{35}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 35:

$\frac{315 \div 35}{35 \div 35} = \frac{9}{1} = 9$.

So, the final result is 9.

Conclusion

By following the order of operations and carefully multiplying and dividing the fractions, we found that $(-4) \times (\frac{7}{5}) \times (-\frac{3}{4}) \div (\frac{7}{15}) = 9$. This exercise demonstrates the importance of understanding basic arithmetic operations and fraction manipulation. Remember to always pay attention to signs and simplify fractions whenever possible to make calculations easier. Practice these types of problems regularly to build confidence and proficiency.

To further enhance your understanding of fraction operations, you might find helpful resources on websites like Khan Academy. They offer comprehensive lessons and practice exercises covering a wide range of math topics.