Solving The Equation 2/5-15+8-1/5+3 A Step-by-Step Guide

Mathematics often presents us with a variety of expressions that require careful manipulation and understanding of fundamental principles. This article delves into a specific mathematical expression: ${\frac{2}{5}-15+8-\frac{1}{5}+3}$. We will break down each step, providing a clear and concise explanation to ensure a thorough comprehension of the solution. Whether you are a student seeking to enhance your mathematical skills or simply curious about problem-solving strategies, this guide offers a valuable resource for navigating such expressions. By focusing on the order of operations, fraction manipulation, and integer arithmetic, we aim to equip you with the tools necessary to tackle similar mathematical challenges with confidence and precision.

Breaking Down the Expression: ${\frac{2}{5}-15+8-\frac{1}{5}+3}$

To effectively solve the mathematical expression ${\frac{2}{5}-15+8-\frac{1}{5}+3}$, it is crucial to understand the order of operations and how to manipulate both fractions and integers. This initial breakdown serves as a foundation for the subsequent steps, ensuring clarity and accuracy in our calculations. The expression involves a combination of fractions and integers, requiring us to address each component systematically. Let’s first identify the key elements: we have two fractions, ${\frac{2}{5}}$ and ${\frac{1}{5}}$, and three integers: -15, +8, and +3. The operations involved are subtraction and addition, which, according to the order of operations (PEMDAS/BODMAS), should be performed from left to right. This means we will first handle the fractions, then the integers, ensuring that each step is logically sequenced. This methodical approach is essential in preventing errors and promoting a clear understanding of the solution process. By carefully dissecting the expression into its constituent parts, we can more easily manage the calculations and arrive at the correct answer. The following sections will detail each step, providing explanations and justifications for every operation performed. This comprehensive approach ensures that readers not only understand the solution but also the underlying principles that govern it.

Step 1: Combining the Fractions

In the first step of solving the expression ${\frac{2}{5}-15+8-\frac{1}{5}+3}$, we focus on combining the fractions. This involves identifying the fractions present in the expression and performing the necessary arithmetic operations to simplify them. The fractions in our expression are ${\frac{2}{5}}$ and ${\frac{1}{5}}$. We need to subtract ${\frac{1}{5}}$ from ${\frac{2}{5}}$. Since the denominators are the same, the operation is straightforward: we simply subtract the numerators while keeping the denominator constant. Mathematically, this can be represented as: ${\frac{2}{5} - \frac{1}{5}}$. To perform this subtraction, we subtract the numerators (2 - 1) and keep the common denominator (5). This yields: ${\frac{2-1}{5} = \frac{1}{5}}$. Thus, the result of combining the fractions ${\frac{2}{5}}$ and ${\frac{1}{5}}$ is ${\frac{1}{5}}$. This simplified fraction will be used in the subsequent steps of solving the overall expression. Combining fractions at the beginning of the process helps streamline the calculations and reduces the complexity of the expression. This step highlights the importance of understanding fraction arithmetic and how to efficiently manipulate fractions within a larger mathematical problem. The next section will focus on combining the integer components of the expression.

Step 2: Combining the Integers

Following the simplification of the fractional components, the next step in solving ${\frac{2}{5}-15+8-\frac{1}{5}+3}$ involves combining the integers. The integers present in the expression are -15, +8, and +3. We will perform the addition and subtraction operations from left to right, as dictated by the order of operations. First, we combine -15 and +8. Adding these integers together, we get: -15 + 8 = -7. This means that the sum of -15 and 8 is -7. Next, we take this result (-7) and add the remaining integer, +3, to it. So, we have: -7 + 3 = -4. Thus, the combined value of the integers -15, +8, and +3 is -4. This simplification is crucial as it reduces the number of terms in the expression, making it easier to manage. Combining integers accurately is a fundamental skill in mathematics, and this step demonstrates its application within a more complex expression. By systematically adding and subtracting the integers, we have reduced the integer portion of the expression to a single value. This allows us to proceed with the final step, which involves combining the simplified fraction and the simplified integer to arrive at the final solution. The following section will detail this final step, bringing together the results of the previous calculations to provide a complete answer.

Step 3: Combining the Simplified Fraction and Integer

After simplifying the fractions and integers separately in the expression ${\frac{2}{5}-15+8-\frac{1}{5}+3}$, the final step involves combining these simplified components to arrive at the ultimate solution. In Step 1, we combined the fractions ${\frac{2}{5}}$ and ${\frac{1}{5}}$ to get ${\frac{1}{5}}$. In Step 2, we combined the integers -15, +8, and +3 to get -4. Now, we need to combine ${\frac{1}{5}}$ and -4. This can be expressed as: ${\frac{1}{5} - 4}$. To combine a fraction and an integer, it is helpful to convert the integer into a fraction with the same denominator as the existing fraction. In this case, we want to express -4 as a fraction with a denominator of 5. To do this, we multiply -4 by ${\frac{5}{5}}$, which is equivalent to 1 and does not change the value of the integer. So, -4 becomes ${\frac{-4 imes 5}{5} = \frac{-20}{5}}$. Now we can rewrite the expression as: ${\frac{1}{5} - \frac{20}{5}}$. Since the denominators are the same, we can subtract the numerators: ${\frac{1 - 20}{5}}$. This gives us: ${\frac{-19}{5}}$. Therefore, the final simplified value of the expression ${\frac{2}{5}-15+8-\frac{1}{5}+3}$ is ${\frac{-19}{5}}$. This result represents the culmination of all the previous steps, demonstrating a systematic approach to solving mathematical expressions involving fractions and integers. The final answer, ${\frac{-19}{5}}$, can also be expressed as a mixed number, which we will explore in the next section.

Expressing the Result as a Mixed Number

The result we obtained, ${\frac{-19}{5}}$, is an improper fraction because the absolute value of the numerator (19) is greater than the denominator (5). To provide a more intuitive understanding of this value, it is often useful to express it as a mixed number. A mixed number combines a whole number and a proper fraction. To convert the improper fraction ${\frac{-19}{5}}$ into a mixed number, we need to divide the numerator (19) by the denominator (5). When we divide 19 by 5, we get a quotient of 3 and a remainder of 4. This means that 5 goes into 19 three times with 4 left over. In terms of the fraction, this tells us that ${\frac{19}{5}}$ can be expressed as 3 whole units and ${\frac{4}{5}}$ of another unit. Since our original fraction was negative, we apply the negative sign to the whole number part of the mixed number. Thus, ${\frac{-19}{5}}$ is equivalent to -3 ${\frac{4}{5}}$. This mixed number representation gives us a clearer sense of the magnitude of the value. It tells us that the number is slightly less than -3 and ${\frac{4}{5}}$. Converting improper fractions to mixed numbers is a valuable skill in mathematics, as it allows for a more practical interpretation of numerical values, especially in contexts where fractions represent quantities or measurements. This conversion provides an alternative way to understand the final result of our expression, further solidifying our comprehension of the solution process. The ability to switch between improper fractions and mixed numbers enhances mathematical fluency and problem-solving capabilities.

Importance of Order of Operations

Throughout the process of solving the expression ${\frac{2}{5}-15+8-\frac{1}{5}+3}$, the order of operations plays a crucial role in ensuring the accuracy and validity of the solution. The order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), provides a set of rules that dictate the sequence in which mathematical operations should be performed. In our expression, we have addition and subtraction involving both fractions and integers. According to the order of operations, we perform these operations from left to right. This means that we first combined the fractions ${\frac{2}{5}}$ and ${\frac{-1}{5}}$, and then we combined the integers -15, 8, and 3, strictly following the left-to-right sequence. Had we deviated from this order, for instance, by adding 8 and -${\frac{1}{5}}$ before combining the fractions, we would likely have arrived at a different and incorrect result. The order of operations is not merely a convention; it is a fundamental principle that ensures mathematical expressions have a consistent and unambiguous interpretation. Without it, the same expression could yield multiple different answers, leading to confusion and errors. The correct application of the order of operations is essential not only in basic arithmetic but also in more advanced mathematical fields such as algebra, calculus, and beyond. By adhering to these rules, we can confidently solve complex mathematical problems and communicate our solutions effectively. This example underscores the critical importance of mastering and consistently applying the order of operations in all mathematical endeavors.

Common Mistakes and How to Avoid Them

When tackling mathematical expressions such as ${\frac{2}{5}-15+8-\frac{1}{5}+3}$, it is crucial to be aware of common mistakes that can lead to incorrect solutions. Identifying these pitfalls and understanding how to avoid them can significantly improve accuracy and confidence in problem-solving. One common mistake is neglecting the order of operations. As discussed earlier, PEMDAS/BODMAS dictates the sequence of operations, and deviating from this order can result in errors. For instance, someone might incorrectly add 8 to -${\frac{1}{5}}$ before combining the fractions ${\frac{2}{5}}$ and ${\frac{-1}{5}}$, leading to a flawed result. To avoid this, always ensure you follow the correct order: simplify fractions, then integers, and perform operations from left to right. Another frequent error involves incorrect fraction manipulation. When adding or subtracting fractions, it is essential to have a common denominator. A mistake could occur if someone attempts to subtract ${\frac{1}{5}}$ directly from 15 without properly converting 15 into a fraction with a denominator of 5. To prevent this, always find a common denominator before performing addition or subtraction with fractions. Sign errors are also common, particularly when dealing with negative numbers. For example, mishandling the negative sign when combining integers can lead to an incorrect sum. To minimize sign errors, it is helpful to rewrite the expression, clearly showing the positive and negative signs, and to double-check each step. Additionally, making arithmetic errors during basic operations such as addition, subtraction, multiplication, or division can derail the entire solution. To avoid such errors, it is beneficial to perform calculations carefully and, if possible, verify your work using a calculator or by working through the problem a second time. By being mindful of these common mistakes and actively working to prevent them, you can enhance your mathematical proficiency and accuracy in solving complex expressions.

Conclusion: Mastering Mathematical Expressions

In conclusion, solving the mathematical expression ${\frac{2}{5}-15+8-\frac{1}{5}+3}$ involves a systematic approach that combines an understanding of fraction arithmetic, integer manipulation, and the crucial order of operations. By breaking down the expression into manageable steps, we can simplify and solve it accurately. We began by combining the fractions, then the integers, and finally combined the simplified fraction and integer to arrive at the solution ${\frac{-19}{5}}$, which can also be expressed as the mixed number -3 ${\frac{4}{5}}$. Throughout this process, we emphasized the importance of adhering to the order of operations to ensure a correct and unambiguous result. We also highlighted common mistakes, such as neglecting the order of operations, improper fraction manipulation, and sign errors, and provided strategies for avoiding them. Mastering the solution of mathematical expressions like this is fundamental to developing strong mathematical skills. It not only enhances your ability to solve specific problems but also cultivates a methodical and precise approach to problem-solving in general. The skills acquired through this process are transferable to various mathematical contexts, from basic arithmetic to more advanced topics. By practicing and understanding the principles outlined in this guide, you can build confidence in your mathematical abilities and effectively tackle a wide range of mathematical challenges. This comprehensive approach empowers you to not only find the correct answer but also to understand the underlying concepts, fostering a deeper appreciation for mathematics.