Solving Complex Math -4 1/7 + 2 1/4 * (-11 2/9 - (-5.4) : 9/35) Step By Step
In the realm of mathematics, expressions that intertwine fractions, decimals, and various operations can often seem like intricate puzzles. This article aims to dissect one such mathematical expression: -4 1/7 + 2 1/4 * (-11 2/9 - (-5.4) : 9/35). By meticulously breaking down each step, we will not only arrive at the solution but also illuminate the underlying principles that govern these calculations. This comprehensive guide will serve as a valuable resource for anyone seeking to enhance their understanding of arithmetic operations involving fractions and decimals.
Unveiling the Order of Operations
Before we embark on the journey of solving this expression, it's paramount to understand the order of operations. This fundamental principle, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed. Adhering to this order is crucial for achieving the correct result. In our case, we will first tackle the operations within the parentheses, followed by multiplication and division, and finally, addition and subtraction. Understanding and applying PEMDAS is the cornerstone of accurate mathematical calculations, particularly when dealing with complex expressions involving multiple operations.
Step 1: Taming the Parentheses
Our initial focus lies within the parentheses: (-11 2/9 - (-5.4) : 9/35). This segment itself comprises a subtraction and a division operation. According to PEMDAS, we must first address the division. However, before we can divide, we need to convert the mixed number (-11 2/9) and the decimal (-5.4) into improper fractions. This conversion is essential because dividing fractions requires both numbers to be in fractional form. Converting mixed numbers to improper fractions involves multiplying the whole number by the denominator and adding the numerator, then placing the result over the original denominator. Converting decimals to fractions involves expressing the decimal as a fraction with a power of 10 in the denominator, then simplifying if possible. This step highlights the importance of understanding different number forms and how to seamlessly transition between them.
Converting to Improper Fractions
Let's begin by converting -11 2/9 into an improper fraction. We multiply the whole number (-11) by the denominator (9) and add the numerator (2), which gives us -99 + 2 = -97. We then place this result over the original denominator, yielding -97/9. This conversion transforms the mixed number into a single fraction, making it easier to perform mathematical operations.
Next, we convert the decimal -5.4 into an improper fraction. We can express -5.4 as -54/10. Simplifying this fraction by dividing both numerator and denominator by their greatest common divisor (2) gives us -27/5. This conversion ensures that both numbers within the parentheses are now in fractional form, paving the way for division.
Performing the Division
Now that we have -27/5, we can proceed with the division: (-27/5) : (9/35). Dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of 9/35 is 35/9. Therefore, the division becomes a multiplication: (-27/5) * (35/9). To multiply fractions, we multiply the numerators and the denominators: (-27 * 35) / (5 * 9) = -945/45. This multiplication transforms the division problem into a more manageable form.
We can simplify the resulting fraction -945/45 by dividing both the numerator and the denominator by their greatest common divisor (45), which gives us -21. This simplification reduces the fraction to its simplest form, making subsequent calculations easier.
Completing the Parenthetical Operation
Now we have -97/9 - (-21). Subtracting a negative number is the same as adding its positive counterpart, so the expression becomes -97/9 + 21. To add these numbers, we need a common denominator. We can express 21 as a fraction with a denominator of 9 by multiplying both the numerator and denominator by 9, which gives us 189/9. Now we can add the fractions: -97/9 + 189/9 = 92/9. We have successfully simplified the expression within the parentheses to a single fraction.
Step 2: Tackling Multiplication
With the parentheses resolved, we move on to the multiplication operation: 2 1/4 * (92/9). Similar to our approach within the parentheses, we first need to convert the mixed number 2 1/4 into an improper fraction. Multiplying the whole number (2) by the denominator (4) and adding the numerator (1) gives us 8 + 1 = 9. Placing this result over the original denominator, we get 9/4. Now we can rewrite the multiplication as (9/4) * (92/9). This conversion ensures that both numbers are in fractional form, ready for multiplication.
To multiply the fractions, we multiply the numerators and the denominators: (9 * 92) / (4 * 9) = 828/36. This multiplication yields a new fraction that we can simplify. We can simplify the fraction 828/36 by dividing both the numerator and the denominator by their greatest common divisor (36), which gives us 23. The multiplication operation has been simplified to a single whole number.
Step 3: The Final Addition
We are now left with the final operation: -4 1/7 + 23. Again, we need to convert the mixed number -4 1/7 into an improper fraction. Multiplying the whole number (-4) by the denominator (7) and adding the numerator (1) gives us -28 + 1 = -29. Placing this result over the original denominator, we get -29/7. Now the expression becomes -29/7 + 23. This conversion prepares the mixed number for addition with the whole number.
To add these numbers, we need a common denominator. We can express 23 as a fraction with a denominator of 7 by multiplying both the numerator and the denominator by 7, which gives us 161/7. Now we can add the fractions: -29/7 + 161/7 = 132/7. This addition combines the two terms into a single fraction.
Converting Back to a Mixed Number (Optional)
While 132/7 is a perfectly valid answer, we can convert it back to a mixed number for a more intuitive representation. To do this, we divide the numerator (132) by the denominator (7), which gives us 18 with a remainder of 6. The quotient (18) becomes the whole number part of the mixed number, the remainder (6) becomes the numerator, and the denominator (7) remains the same. Therefore, 132/7 is equivalent to 18 6/7. This final conversion provides an alternative representation of the solution.
Conclusion: Triumph Over Complexity
By meticulously following the order of operations and employing the principles of fraction and decimal arithmetic, we have successfully navigated the complex expression -4 1/7 + 2 1/4 * (-11 2/9 - (-5.4) : 9/35). The solution, 132/7 or 18 6/7, stands as a testament to the power of systematic problem-solving. This exercise underscores the importance of mastering fundamental mathematical concepts and applying them with precision and care. The ability to dissect complex expressions into manageable steps is a valuable skill that transcends the realm of mathematics, fostering critical thinking and analytical prowess in various aspects of life.
