Calvin's Calculation Error Analysis Of -2/3 * 5 1/6

This article delves into a mathematical problem encountered by Calvin while evaluating the expression $-\frac{2}{3} \times 5 \frac{1}{6}$. We will meticulously examine each step of Calvin's solution, pinpoint the error, and provide a correct solution along with a comprehensive explanation. This analysis aims to enhance understanding of fraction multiplication and the distributive property, crucial concepts in mathematics.

Understanding the Problem

Before diving into Calvin's steps, let's clearly define the problem. Calvin was tasked with calculating the product of a negative fraction, $-\frac{2}{3}$, and a mixed number, $5 \frac{1}{6}$. To solve this accurately, it's essential to remember the rules of fraction multiplication and how to convert mixed numbers into improper fractions. The correct approach involves converting the mixed number into an improper fraction and then multiplying the numerators and denominators accordingly. This foundational understanding is key to avoiding common pitfalls in mathematical calculations.

Calvin's Attempted Solution

To identify where the mistake occurred, let's break down Calvin's attempted solution step by step. This methodical approach allows us to pinpoint the exact location of the error and understand the underlying misconception. Analyzing each step helps to reinforce the correct mathematical procedures and highlight common mistakes to avoid. The original problem is:

$-\frac{2}{3} \times 5 \frac{1}{6}$

Step 1: $-\frac{2}{3} \times\left(5+\frac{1}{6}\right)$

In Step 1, Calvin correctly rewrites the mixed number $5 \frac{1}{6}$ as the sum of its whole number and fractional parts, which is $(5 + \frac{1}{6})$. This is a valid initial step, demonstrating an understanding of how mixed numbers are composed. This step sets the stage for potentially using the distributive property, but it's crucial to apply it correctly in the subsequent steps. The equivalence of a mixed number and the sum of its parts is a fundamental concept in arithmetic, and Calvin's correct application here shows a solid grasp of this principle.

Step 2: $\left(-\frac{2}{3}+5\right) \times\left(-\frac{2}{3}+\frac{1}{6}\right)$

Step 2 is where Calvin's critical error occurs. Instead of correctly applying the distributive property, which would involve multiplying $-\frac{2}{3}$ by both 5 and $\frac{1}{6}$, Calvin incorrectly adds $-\frac{2}{3}$ to both 5 and $\frac{1}{6}$. This is a fundamental misunderstanding of the distributive property, which states that a(b + c) = ab + ac. This error leads to an incorrect result and highlights the importance of understanding and correctly applying mathematical properties. Recognizing this error is crucial for learning and avoiding similar mistakes in the future. The distributive property is a cornerstone of algebraic manipulation, and its correct application is essential for solving a wide range of mathematical problems. This step underscores the necessity of careful attention to detail and a solid understanding of mathematical principles.

Step 3: $4 \frac{1}{3}$

Step 3 presents the final, incorrect answer of $4 \frac{1}{3}$. This result stems directly from the error made in Step 2. Because the distributive property was misapplied, the subsequent calculation leads to a wrong conclusion. While the arithmetic performed in this step might be correct based on the incorrect values from Step 2, the final answer is invalid due to the initial error. This highlights the cascading effect of errors in mathematical calculations; a single mistake early in the process can invalidate the entire solution. Therefore, it's crucial to verify each step and ensure the correct application of mathematical principles. This incorrect answer serves as a stark reminder of the importance of accuracy and precision in mathematical problem-solving. It also emphasizes the value of checking one's work to identify and correct errors before arriving at a final answer.

Correct Solution

To arrive at the correct solution, we need to follow the proper steps for multiplying a fraction by a mixed number. This involves converting the mixed number to an improper fraction and then performing the multiplication. Understanding the correct procedure is essential for obtaining the accurate answer and reinforcing mathematical skills. Let's walk through the steps to ensure clarity and comprehension.

Step 1: Convert the Mixed Number to an Improper Fraction

First, we convert the mixed number $5 \frac{1}{6}$ into an improper fraction. To do this, we multiply the whole number (5) by the denominator (6) and add the numerator (1). This result becomes the new numerator, and the denominator remains the same. So, $5 \frac{1}{6} = \frac{(5 \times 6) + 1}{6} = \frac{31}{6}$. Converting mixed numbers to improper fractions is a crucial step in many fraction-related calculations, as it simplifies the multiplication and division processes. This conversion allows us to work with a single fraction instead of a combination of a whole number and a fraction, making the subsequent calculations more straightforward. Mastery of this conversion is fundamental for success in more advanced mathematical topics.

Step 2: Multiply the Fractions

Now, we multiply the fractions $-\frac{2}{3}$ and $\frac{31}{6}$. To multiply fractions, we multiply the numerators together and the denominators together: $-\frac{2}{3} \times \frac{31}{6} = -\frac{2 \times 31}{3 \times 6} = -\frac{62}{18}$. This step demonstrates the fundamental rule of fraction multiplication, which is to multiply the top numbers (numerators) and the bottom numbers (denominators). The negative sign is carried through the calculation, ensuring the correct sign in the final answer. Understanding and applying this rule is essential for performing fraction multiplications accurately. The resulting fraction, $-\frac{62}{18}$, is an improper fraction that can be simplified further.

Step 3: Simplify the Result

Finally, we simplify the improper fraction $-\frac{62}{18}$. Both 62 and 18 are divisible by 2, so we can reduce the fraction: $-\frac{62}{18} = -\frac{31}{9}$. Now, we convert the improper fraction $-\frac{31}{9}$ back to a mixed number. We divide 31 by 9, which gives us 3 with a remainder of 4. Therefore, $-\frac{31}{9} = -3 \frac{4}{9}$. Simplifying fractions is a crucial step in mathematical problem-solving, as it presents the answer in its most concise and understandable form. Converting improper fractions back to mixed numbers is often preferred, especially in practical applications, as it provides a clearer sense of the quantity. This final step ensures that the answer is presented in the most appropriate and easily interpretable format. Therefore, the correct answer is $-3 \frac{4}{9}$.

Key Takeaways and Learning Points

This problem highlights several key takeaways for mathematical understanding. First and foremost, it emphasizes the critical importance of correctly applying the distributive property. Misunderstanding or misapplying this property can lead to significant errors in calculations. The distributive property is a fundamental concept in algebra and is used extensively in simplifying expressions and solving equations. Therefore, a thorough understanding of its application is essential for mathematical proficiency. This example serves as a valuable reminder to double-check the application of mathematical properties in problem-solving. Another key learning point is the necessity of understanding fraction operations, including converting mixed numbers to improper fractions and simplifying the results. These skills are foundational for more advanced mathematical concepts. Mastering fraction operations not only helps in solving problems involving fractions but also builds a strong base for algebra, calculus, and other higher-level mathematics. Consistent practice and a solid grasp of these fundamentals are crucial for success in mathematics. Finally, this problem underscores the importance of a step-by-step approach to problem-solving. Breaking down the problem into smaller, manageable steps allows for easier identification of errors and a clearer understanding of the solution process. This methodical approach is not only beneficial in mathematics but also in various other problem-solving situations. By carefully analyzing each step, we can ensure accuracy and avoid common pitfalls. This problem provides a valuable learning opportunity for students to strengthen their understanding of mathematical principles and problem-solving strategies.

Conclusion

In conclusion, Calvin's error in evaluating $-\frac{2}{3} \times 5 \frac{1}{6}$ stemmed from a misapplication of the distributive property. By correctly converting the mixed number to an improper fraction and following the rules of fraction multiplication, we arrived at the correct answer of $-3 \frac{4}{9}$. This exercise reinforces the importance of understanding fundamental mathematical principles and applying them accurately. It also highlights the value of a step-by-step approach to problem-solving, which allows for the identification and correction of errors. By learning from mistakes and reinforcing correct procedures, students can develop a stronger foundation in mathematics and improve their problem-solving skills. This detailed analysis of Calvin's error provides a valuable learning opportunity for anyone studying or practicing mathematical concepts. The key is to recognize the error, understand the correct method, and apply the knowledge to future problems. With practice and a solid understanding of mathematical principles, anyone can improve their problem-solving abilities and achieve success in mathematics.