Solving (2 6/7 - 2.8) ÷ (1 2/3 - 1.6) A Step-by-Step Guide

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Introduction

In the realm of mathematics, we often encounter expressions that require us to perform a series of operations to arrive at the final answer. One such expression is (2 6/7 - 2.8) ÷ (1 2/3 - 1.6). This expression involves fractions, decimals, and the order of operations, making it a fascinating problem to solve. In this comprehensive guide, we will embark on a step-by-step journey to dissect this expression, understand the underlying mathematical principles, and ultimately arrive at the solution. We will explore the individual components of the expression, convert fractions to decimals, perform subtraction operations, and finally, divide the results to obtain the final answer. By the end of this guide, you will have a clear understanding of how to tackle similar mathematical expressions with confidence and precision.

Step 1: Converting Mixed Numbers to Improper Fractions

The first step in solving this expression is to convert the mixed numbers into improper fractions. This conversion will make it easier to perform the subsequent arithmetic operations. Let's begin by converting the mixed number 2 6/7 into an improper fraction. To do this, we multiply the whole number (2) by the denominator (7) and add the numerator (6). This gives us (2 * 7) + 6 = 14 + 6 = 20. We then place this result over the original denominator, resulting in the improper fraction 20/7. Similarly, we convert the mixed number 1 2/3 into an improper fraction. Multiplying the whole number (1) by the denominator (3) and adding the numerator (2) gives us (1 * 3) + 2 = 3 + 2 = 5. Placing this over the original denominator yields the improper fraction 5/3. Now, our expression looks like this: (20/7 - 2.8) ÷ (5/3 - 1.6). This conversion is crucial as it sets the stage for simplifying the expression further and performing the necessary calculations with ease.

Step 2: Converting Fractions to Decimals

To simplify the expression further, let's convert the fractions into decimals. This will allow us to perform the subtraction operations more easily. We begin by converting the fraction 20/7 into a decimal. Dividing 20 by 7, we get approximately 2.857. Next, we convert the fraction 5/3 into a decimal. Dividing 5 by 3, we get approximately 1.667. Now, our expression looks like this: (2.857 - 2.8) ÷ (1.667 - 1.6). Converting fractions to decimals is a fundamental step in simplifying expressions that involve both fractions and decimals. This conversion allows us to perform arithmetic operations using a consistent number format, making the calculations more straightforward and less prone to errors.

Step 3: Performing Subtraction within Parentheses

Now that we have converted the fractions to decimals, we can perform the subtraction operations within the parentheses. First, let's subtract 2.8 from 2.857. This gives us 0.057. Next, let's subtract 1.6 from 1.667. This gives us 0.067. Our expression now looks like this: 0.057 ÷ 0.067. Performing the subtraction within the parentheses is a crucial step in following the order of operations, which dictates that we must perform operations within parentheses before any other operations. This step simplifies the expression by reducing the number of terms and operations, bringing us closer to the final solution.

Step 4: Performing the Division

The final step is to perform the division operation. We need to divide 0.057 by 0.067. Performing this division, we get approximately 0.851. Therefore, the solution to the expression (2 6/7 - 2.8) ÷ (1 2/3 - 1.6) is approximately 0.851. The division operation is the final step in this mathematical journey, bringing together all the previous steps to arrive at the ultimate answer. By performing this division accurately, we complete the solution and gain a deeper understanding of the mathematical expression.

Alternative Approach: Working with Fractions Throughout

While converting to decimals is a common approach, we can also solve this problem by working with fractions throughout. This alternative method provides a different perspective and reinforces our understanding of fraction operations. Let's revisit the expression: (2 6/7 - 2.8) ÷ (1 2/3 - 1.6). We've already converted the mixed numbers to improper fractions: (20/7 - 2.8) ÷ (5/3 - 1.6). Now, let's convert the decimals to fractions. 2.8 can be written as 28/10, which simplifies to 14/5. 1.6 can be written as 16/10, which simplifies to 8/5. Our expression now becomes: (20/7 - 14/5) ÷ (5/3 - 8/5). To subtract fractions, we need a common denominator. For the first set of parentheses, the least common denominator (LCD) of 7 and 5 is 35. Converting the fractions, we get: (100/35 - 98/35). For the second set of parentheses, the LCD of 3 and 5 is 15. Converting the fractions, we get: (25/15 - 24/15). Performing the subtractions, we have: (2/35) ÷ (1/15). To divide fractions, we multiply by the reciprocal of the divisor: (2/35) * (15/1). This simplifies to 30/35, which further simplifies to 6/7. Converting 6/7 to a decimal gives us approximately 0.857, which is slightly different from our previous answer due to rounding in the decimal method. This alternative approach highlights the importance of understanding fraction operations and provides a valuable method for solving mathematical expressions.

Conclusion

In this comprehensive guide, we have successfully navigated the mathematical expression (2 6/7 - 2.8) ÷ (1 2/3 - 1.6). We explored the step-by-step process of converting mixed numbers to improper fractions, fractions to decimals, performing subtraction operations within parentheses, and finally, dividing the results to obtain the solution. We also examined an alternative approach that involved working with fractions throughout the calculation. The solution we arrived at is approximately 0.851 using the decimal method and approximately 0.857 using the fraction method. The slight difference is due to rounding errors in the decimal conversion. By understanding the order of operations and the principles of fraction and decimal arithmetic, we can confidently tackle similar mathematical challenges. This exercise has not only provided us with a solution to a specific problem but has also enhanced our overall mathematical skills and problem-solving abilities. The ability to break down complex expressions into smaller, manageable steps is a valuable asset in mathematics and in various other fields that require analytical thinking and problem-solving.