Solving The Complex Mathematical Expression (13.75 + 9 * 1/6) * 1.2 / (10.3 - 8 * 1/2) * 5/9 + (6.9 - 3 3/5) * 5 5/6 / (3 2/3 - 3 1/6) * 56 - 27 1/6

Navigating the intricate world of mathematics often involves tackling complex expressions that demand a meticulous step-by-step approach. In this article, we will embark on a journey to unravel the solution to a challenging mathematical expression, breaking it down into manageable segments and elucidating the underlying principles at play. The expression we aim to conquer is: (13.75 + 9 * 1/6) * 1.2 / (10.3 - 8 * 1/2) * 5/9 + (6.9 - 3 3/5) * 5 5/6 / (3 2/3 - 3 1/6) * 56 - 27 1/6. Prepare to delve into the realm of fractions, decimals, and order of operations as we dissect this equation and arrive at the final answer.

Deciphering the Initial Components

To effectively solve this mathematical puzzle, we must first focus on the initial components of the expression. This involves carefully examining each element and understanding its role within the larger equation. Our first step is to dissect the terms within the parentheses, as these often form the building blocks of more complex calculations. The expression begins with (13.75 + 9 * 1/6) * 1.2. Here, we encounter a combination of decimal and fractional values, along with the operations of addition and multiplication. To proceed accurately, we need to adhere to the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This principle dictates that we perform multiplication before addition.

Let's break down the first part, (13.75 + 9 * 1/6). According to the order of operations, we first handle the multiplication: 9 * 1/6. Multiplying 9 by 1/6 is equivalent to dividing 9 by 6, which yields 1.5. Now we have 13.75 + 1.5. Adding these two values together gives us 15.25. This result represents the simplified value of the first set of parentheses. The next step is to multiply this result by 1.2. So, we calculate 15.25 * 1.2. This multiplication results in 18.3. Therefore, the first segment of the expression, (13.75 + 9 * 1/6) * 1.2, simplifies to 18.3. This meticulous approach to breaking down the initial components ensures accuracy and lays the foundation for tackling the rest of the expression.

Tackling the Denominator: A Step-by-Step Analysis

In this section, we focus on the denominator of the first major term in our expression. The denominator is (10.3 - 8 * 1/2). As with the previous section, we must adhere to the order of operations, PEMDAS, which prioritizes multiplication before subtraction. Within the parentheses, we first address the multiplication: 8 * 1/2. Multiplying 8 by 1/2 is equivalent to finding half of 8, which is 4. Now we have the expression 10.3 - 4. Subtracting 4 from 10.3 results in 6.3. Therefore, the denominator (10.3 - 8 * 1/2) simplifies to 6.3.

This step is crucial because it sets the stage for the division operation that follows. Having simplified the numerator in the previous section to 18.3, we can now perform the division 18.3 / 6.3. This division is a key step in unraveling the first major term of the expression. Accurate calculation of the denominator ensures that the subsequent division yields a correct result. Moreover, understanding the order of operations is paramount in this process, as performing subtraction before multiplication would lead to an incorrect answer. By systematically breaking down the denominator and applying the correct mathematical principles, we pave the way for solving the overall expression.

Solving the First Major Term: Division and Multiplication

Now that we've simplified both the numerator and the denominator, our focus shifts to solving the first major term of the expression. We've established that the numerator, (13.75 + 9 * 1/6) * 1.2, simplifies to 18.3, and the denominator, (10.3 - 8 * 1/2), simplifies to 6.3. This means we now need to perform the division 18.3 / 6.3. Dividing 18.3 by 6.3 gives us approximately 2.90476 (rounded to five decimal places).

With this division completed, we move on to the next operation within the first major term, which is multiplication by 5/9. We multiply our result, approximately 2.90476, by 5/9. Multiplying a number by a fraction involves multiplying the number by the numerator of the fraction and then dividing by the denominator. So, we calculate 2.90476 * 5, which equals approximately 14.5238. Next, we divide this result by 9: 14.5238 / 9. This division yields approximately 1.61376 (rounded to five decimal places). Therefore, the first major term of the expression, (13.75 + 9 * 1/6) * 1.2 / (10.3 - 8 * 1/2) * 5/9, simplifies to approximately 1.61376. This step-by-step approach, involving division and multiplication, demonstrates the importance of adhering to the order of operations and performing calculations with precision.

Navigating the Second Term: A Deep Dive

Our focus now shifts to the second major term of the expression: (6.9 - 3 3/5) * 5 5/6 / (3 2/3 - 3 1/6) * 56. This term presents a combination of decimals, mixed numbers, and various arithmetic operations. To tackle this effectively, we will again employ the order of operations (PEMDAS) and break the term down into smaller, more manageable segments.

First, let's address the expression within the first set of parentheses: (6.9 - 3 3/5). We need to convert the mixed number 3 3/5 into a decimal or an improper fraction for easier calculation. Converting 3 3/5 to an improper fraction, we get (3 * 5 + 3) / 5 = 18/5. As a decimal, 18/5 is 3.6. Now we can subtract this from 6.9: 6.9 - 3.6 = 3.3. So, the first set of parentheses simplifies to 3.3.

Next, we focus on the second set of parentheses: 5 5/6. Converting this mixed number to an improper fraction, we get (5 * 6 + 5) / 6 = 35/6. This value will be used later in the multiplication and division steps. Then, we move to the third set of parentheses: (3 2/3 - 3 1/6). Converting these mixed numbers to improper fractions, we have 3 2/3 = (3 * 3 + 2) / 3 = 11/3 and 3 1/6 = (3 * 6 + 1) / 6 = 19/6. To subtract these fractions, we need a common denominator, which is 6. Converting 11/3 to have a denominator of 6, we get 22/6. Now we can subtract: 22/6 - 19/6 = 3/6, which simplifies to 1/2 or 0.5. Therefore, the third set of parentheses simplifies to 0.5. This thorough analysis of the components within the second term lays the groundwork for the subsequent calculations.

Unraveling the Second Term: Multiplication, Division, and a Final Flourish

Having dissected the components of the second term, our focus now turns to performing the operations. We have simplified (6.9 - 3 3/5) to 3.3, 5 5/6 to 35/6, and (3 2/3 - 3 1/6) to 0.5. The second term now looks like this: 3.3 * 35/6 / 0.5 * 56. Following the order of operations, we perform multiplication and division from left to right.

First, we multiply 3.3 by 35/6. Converting 3.3 to a fraction, we get 33/10. So, we have (33/10) * (35/6). Multiplying the numerators and denominators, we get (33 * 35) / (10 * 6) = 1155/60. Simplifying this fraction, we get 19.25. Next, we divide this result by 0.5: 19.25 / 0.5 = 38.5. Finally, we multiply this by 56: 38.5 * 56 = 2156. Therefore, the second major term of the expression, (6.9 - 3 3/5) * 5 5/6 / (3 2/3 - 3 1/6) * 56, simplifies to 2156.

This sequence of multiplication and division underscores the importance of adhering to the order of operations. By meticulously performing each step, we arrive at the simplified value of the second term. This result will be crucial in the final stage of our calculation, where we combine the values of the two major terms to arrive at the ultimate solution.

The Grand Finale: Combining Terms and Reaching the Solution

Having diligently simplified both major terms of the expression, we now arrive at the final stage: combining these terms to reach the ultimate solution. We've established that the first major term, (13.75 + 9 * 1/6) * 1.2 / (10.3 - 8 * 1/2) * 5/9, simplifies to approximately 1.61376, and the second major term, (6.9 - 3 3/5) * 5 5/6 / (3 2/3 - 3 1/6) * 56, simplifies to 2156. The original expression also includes a subtraction of 27 1/6 at the end.

So, we now have the expression 1.61376 + 2156 - 27 1/6. First, let's convert the mixed number 27 1/6 to a decimal or an improper fraction. As an improper fraction, 27 1/6 is (27 * 6 + 1) / 6 = 163/6. As a decimal, this is approximately 27.16667. Now we can perform the addition and subtraction: 1. 61376 + 2156 - 27.16667. Adding 1.61376 to 2156 gives us 2157.61376. Then, subtracting 27.16667 from this result gives us approximately 2130.44709.

Therefore, the final solution to the complex mathematical expression (13.75 + 9 * 1/6) * 1.2 / (10.3 - 8 * 1/2) * 5/9 + (6.9 - 3 3/5) * 5 5/6 / (3 2/3 - 3 1/6) * 56 - 27 1/6 is approximately 2130.44709. This journey through the intricate equation highlights the importance of meticulous calculation, adherence to the order of operations, and the power of breaking down complex problems into smaller, more manageable steps. By systematically addressing each component, we have successfully navigated the expression and arrived at the final answer.

In conclusion, solving complex mathematical expressions requires a combination of understanding mathematical principles, meticulous calculation, and a systematic approach. By breaking down the expression into smaller parts, adhering to the order of operations, and carefully performing each calculation, we can successfully unravel even the most challenging equations. The solution to the expression (13.75 + 9 * 1/6) * 1.2 / (10.3 - 8 * 1/2) * 5/9 + (6.9 - 3 3/5) * 5 5/6 / (3 2/3 - 3 1/6) * 56 - 27 1/6 is approximately 2130.44709, a testament to the power of methodical problem-solving in mathematics.