Solving Fraction Expressions A Step-by-Step Guide

Navigating the world of fractions can sometimes feel like traversing a complex maze. However, with a systematic approach and a clear understanding of the rules, even seemingly daunting expressions can be tamed. This comprehensive guide delves into the step-by-step solution of the fractional expression ${\frac{2}{5} \times \frac{-3}{7}-\frac{1}{14}-\frac{3}{7} \times \frac{3}{5}}$, providing not just the answer but also a thorough explanation of the underlying principles. Whether you're a student grappling with homework or an adult seeking to refresh your math skills, this article will equip you with the knowledge and confidence to conquer similar challenges. We'll break down each operation, emphasizing the importance of order of operations and the nuances of dealing with negative fractions. So, let's embark on this mathematical journey together and unlock the secrets of fraction manipulation.

Understanding the Problem: A Step-by-Step Breakdown

Before we dive into the solution, let's carefully dissect the problem at hand: ${\frac{2}{5} \times \frac{-3}{7}-\frac{1}{14}-\frac{3}{7} \times \frac{3}{5}}$. This expression involves a combination of multiplication and subtraction of fractions. To solve it accurately, we must adhere to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In this case, we'll focus on multiplication first, followed by subtraction. Understanding this hierarchy is crucial for arriving at the correct answer. We will begin by addressing the multiplication operations present in the expression. There are two multiplication operations that need to be performed before we can tackle the subtraction. These operations involve multiplying fractions, which requires a specific procedure. By isolating and solving these multiplications first, we simplify the overall expression and make it easier to manage. Let's delve deeper into the rules of fraction multiplication and then apply them to our specific problem. Remember, a clear understanding of each step is paramount to mastering fraction operations. This initial breakdown sets the stage for a smoother, more confident problem-solving experience.

Step 1: Multiplying Fractions – The Foundation of Our Solution

The first key step in simplifying the expression ${\frac{2}{5} \times \frac{-3}{7}-\frac{1}{14}-\frac{3}{7} \times \frac{3}{5}}$ involves understanding how to multiply fractions. The rule is straightforward: multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. Importantly, when multiplying fractions, there is no need to find a common denominator, a common requirement for addition and subtraction. This direct multiplication makes this operation relatively simple. Let's apply this rule to the first multiplication in our expression: ${\frac{2}{5} \times \frac{-3}{7}}$. Multiplying the numerators, we get 2 * (-3) = -6. Multiplying the denominators, we get 5 * 7 = 35. Therefore, ${\frac{2}{5} \times \frac{-3}{7} = \frac{-6}{35}}$. It's crucial to remember the rules of multiplying signed numbers: a positive number multiplied by a negative number results in a negative number. This principle is fundamental to correctly handling the negative fraction in our expression. Now, let's move on to the second multiplication in the expression. By mastering this simple multiplication rule, you lay the groundwork for tackling more complex fractional expressions with confidence. Accuracy in this step is paramount, as it directly impacts the subsequent steps and the final solution.

Step 2: Tackling the Second Multiplication

Having successfully handled the first multiplication, we now turn our attention to the second multiplication in the expression: ${\frac{3}{7} \times \frac{3}{5}}$. Again, we apply the fundamental rule of multiplying fractions: multiply the numerators and multiply the denominators. In this case, we have 3 multiplied by 3 for the numerators, resulting in 9, and 7 multiplied by 5 for the denominators, resulting in 35. Thus, ${\frac{3}{7} \times \frac{3}{5} = \frac{9}{35}}$. This calculation is relatively straightforward, but it's important to execute it accurately. Now, let's pause and take stock of our progress. We have successfully performed both multiplication operations in the original expression. This simplification is key to making the remaining steps, which involve subtraction, much more manageable. We now have a clearer picture of the problem, which is a series of subtractions involving fractions. With the multiplications out of the way, we can focus on the next set of operations. This methodical approach, tackling one operation at a time, is a powerful strategy for solving complex mathematical problems. It reduces the chance of errors and helps to maintain clarity throughout the process. Before proceeding to the subtraction steps, we will rewrite the expression with the results of our multiplications.

Simplifying the Expression: Subtraction Steps

Now that we've tackled the multiplication aspects of the expression ${\frac{2}{5} \times \frac{-3}{7}-\frac{1}{14}-\frac{3}{7} \times \frac{3}{5}}$, let's move on to the subtraction. Our expression, after performing the multiplications, now looks like this: ${\frac{-6}{35} - \frac{1}{14} - \frac{9}{35}}$. To subtract fractions, a crucial requirement is that they must have a common denominator. This common denominator allows us to subtract the numerators while keeping the denominator consistent. If the fractions do not initially have a common denominator, we must find the least common multiple (LCM) of the denominators and convert each fraction accordingly. In our case, the denominators are 35 and 14. To find the LCM, we can list the multiples of each number or use prime factorization. The multiples of 35 are 35, 70, 105, and so on. The multiples of 14 are 14, 28, 42, 56, 70, and so on. We can see that the least common multiple of 35 and 14 is 70. Therefore, we need to convert each fraction to have a denominator of 70. This conversion is a critical step in ensuring accurate subtraction. We will now demonstrate how to convert each fraction to its equivalent form with the common denominator of 70. Mastering this process is essential for confidently handling fraction subtraction problems. Remember, accuracy in finding the LCM and converting fractions is paramount to arriving at the correct solution.

Step 3: Finding the Least Common Denominator (LCD)

Before we can subtract the fractions in our expression ${\frac{-6}{35} - \frac{1}{14} - \frac{9}{35}}$, we must first find the least common denominator (LCD). As previously discussed, the denominators we are working with are 35 and 14. The LCD is the smallest number that is a multiple of both denominators. We've already established that the LCM of 35 and 14 is 70. This means that 70 is our LCD. Identifying the LCD is a fundamental step in adding or subtracting fractions, as it allows us to express each fraction with a common base, making the subtraction operation possible. There are different methods for finding the LCD, including listing multiples and using prime factorization. Regardless of the method used, the goal remains the same: to find the smallest number divisible by all denominators. With our LCD of 70 in hand, we are now ready to convert each fraction in the expression to an equivalent fraction with this denominator. This involves multiplying both the numerator and the denominator of each fraction by a suitable factor that will result in the desired denominator. This process of converting fractions is a key skill in fraction arithmetic. Accurately determining the LCD is crucial because an incorrect LCD will lead to incorrect subtraction and an incorrect final answer. Now that we have the LCD, let's move on to the next step: converting the fractions.

Step 4: Converting Fractions to Equivalent Forms

With the least common denominator (LCD) of 70 determined, we now need to convert each fraction in the expression ${\frac{-6}{35} - \frac{1}{14} - \frac{9}{35}}$ to an equivalent fraction with a denominator of 70. To convert ${\frac{-6}{35}}$ to an equivalent fraction with a denominator of 70, we need to determine what number to multiply 35 by to get 70. Since 35 multiplied by 2 equals 70, we multiply both the numerator and the denominator of ${\frac{-6}{35}}$ by 2. This gives us ${\frac{-6 imes 2}{35 imes 2} = \frac{-12}{70}}$. Next, we convert ${\frac{1}{14}}$ to an equivalent fraction with a denominator of 70. We need to determine what number to multiply 14 by to get 70. Since 14 multiplied by 5 equals 70, we multiply both the numerator and the denominator of ${\frac{1}{14}}$ by 5. This gives us ${\frac{1 imes 5}{14 imes 5} = \frac{5}{70}}$. Finally, we convert ${\frac{9}{35}}$ to an equivalent fraction with a denominator of 70. As we determined earlier, we multiply both the numerator and the denominator of ${\frac{9}{35}}$ by 2. This gives us ${\frac{9 imes 2}{35 imes 2} = \frac{18}{70}}$. By converting each fraction to an equivalent form with the common denominator of 70, we have set the stage for performing the subtraction operations. This step is essential because it ensures that we are subtracting fractions with comparable denominators, which is a fundamental rule of fraction arithmetic. We will now rewrite the expression with the converted fractions and proceed with the subtraction.

Step 5: Performing the Subtraction

Having successfully converted all fractions to have a common denominator, our expression now reads: ${\frac{-12}{70} - \frac{5}{70} - \frac{18}{70}}$. With a common denominator, we can proceed with the subtraction by focusing on the numerators. We subtract the numerators in the order they appear in the expression. First, we subtract 5 from -12, which gives us -17. So, ${\frac{-12}{70} - \frac{5}{70} = \frac{-17}{70}}$. Next, we subtract 18 from -17, which gives us -35. Therefore, ${\frac{-17}{70} - \frac{18}{70} = \frac{-35}{70}}$. So far, we've diligently applied the rules of fraction subtraction, arriving at an intermediate result. However, the journey isn't quite over yet. It's crucial to always check if the resulting fraction can be simplified further. Simplification not only presents the answer in its most concise form but also demonstrates a thorough understanding of fraction manipulation. The fraction we've obtained, ${\frac{-35}{70}}$, certainly looks like it might be simplifiable. Let's move on to the final step, where we'll explore how to reduce this fraction to its simplest form. Remember, simplifying fractions is a vital skill in mathematics, ensuring that answers are presented in the most elegant and practical way.

Finalizing the Solution: Simplification

We've reached the final stage of solving our expression: simplification. After performing the subtractions, we arrived at the fraction ${\frac{-35}{70}}$. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both by that GCD. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. In our case, the numerator is -35 and the denominator is 70. The factors of 35 are 1, 5, 7, and 35. The factors of 70 are 1, 2, 5, 7, 10, 14, 35, and 70. The greatest common divisor of 35 and 70 is 35. Therefore, to simplify the fraction ${\frac{-35}{70}}$, we divide both the numerator and the denominator by 35. This gives us (\frac{-35

\div 35}{70 \div 35} = \frac{-1}{2}). Thus, the simplified form of ${\frac{-35}{70}}$ is ${\frac{-1}{2}}$. This is our final answer. By simplifying the fraction, we have presented the solution in its most concise and understandable form. Simplification is a crucial step in any fraction problem, as it ensures that the answer is in its simplest terms. It also demonstrates a strong understanding of fraction manipulation. We have now successfully navigated through all the steps, from understanding the problem to simplifying the final result. The final simplified answer to the expression ${\frac{2}{5} \times \frac{-3}{7}-\frac{1}{14}-\frac{3}{7} \times \frac{3}{5}}$ is ${\frac{-1}{2}}$.

Step 6: Simplifying the Fraction for the Final Answer

Now, let's simplify the fraction ${\frac{-35}{70}}$ to its simplest form. To do this, we need to find the greatest common divisor (GCD) of the numerator (-35) and the denominator (70). The GCD is the largest number that divides both numbers evenly. By observation, we can see that both -35 and 70 are divisible by 35. Therefore, 35 is the GCD. To simplify, we divide both the numerator and the denominator by the GCD: ${\frac{-35 \div 35}{70 \div 35}}$. Performing the division, we get ${\frac{-1}{2}}$. This fraction is in its simplest form, as -1 and 2 have no common factors other than 1. Therefore, the final simplified answer to the expression ${\frac{2}{5} \times \frac{-3}{7}-\frac{1}{14}-\frac{3}{7} \times \frac{3}{5}}$ is ${\frac{-1}{2}}$. This entire process, from the initial expression to the final simplified answer, demonstrates the importance of following the order of operations, finding common denominators, and simplifying fractions. These are fundamental skills in mathematics and are essential for success in algebra and beyond. By mastering these concepts, you can confidently tackle more complex mathematical problems. Congratulations on reaching the solution! The journey through this problem has hopefully solidified your understanding of fraction operations.

Conclusion: The Final Result and Key Takeaways

In conclusion, we have successfully navigated the complexities of the expression ${\frac{2}{5} \times \frac{-3}{7}-\frac{1}{14}-\frac{3}{7} \times \frac{3}{5}}$ and arrived at the simplified answer of ${\frac{-1}{2}}$. This journey involved several key steps, including understanding the order of operations, multiplying fractions, finding the least common denominator, converting fractions to equivalent forms, performing subtraction, and simplifying the final result. Each step is crucial for accuracy, and mastering these techniques will build a strong foundation for more advanced mathematical concepts. The order of operations, often remembered as PEMDAS, ensures that we perform calculations in the correct sequence, preventing errors. Finding the least common denominator is essential for adding and subtracting fractions, as it allows us to work with comparable units. Simplifying fractions is the final touch, presenting the answer in its most concise and understandable form. Beyond the specific solution, this exercise highlights the importance of a methodical approach to problem-solving. By breaking down a complex expression into smaller, manageable steps, we can tackle even the most daunting mathematical challenges. The key takeaways from this exercise extend beyond the specific problem itself. They encompass the broader principles of mathematical thinking: accuracy, attention to detail, and a systematic approach. By applying these principles, you can confidently approach any mathematical problem, no matter how complex. This comprehensive guide has provided not just the answer but also a pathway to understanding. We encourage you to practice similar problems, reinforcing your skills and building your confidence in the world of fractions. The ability to manipulate fractions is a valuable skill, not only in mathematics but also in various real-world applications.