Step-by-Step Solution To -9/11 + 2/3 + (-3/4)

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In this article, we will walk through the process of solving the mathematical expression -9/11 + 2/3 + (-3/4). This involves adding and subtracting fractions, a fundamental concept in mathematics. We'll break down each step, ensuring clarity and understanding. Whether you're a student looking to improve your math skills or simply curious about how to solve such problems, this guide will provide you with a comprehensive solution.

Understanding the Problem

The problem at hand is to compute the sum of three fractions: -9/11, 2/3, and -3/4. To effectively add or subtract fractions, they must have a common denominator. This means we need to find a number that is a multiple of all the denominators (11, 3, and 4). The least common multiple (LCM) is the smallest such number, and it simplifies the calculations. Once we have a common denominator, we can add or subtract the numerators while keeping the denominator the same. Let’s dive into the steps to find the solution.

Finding the Least Common Multiple (LCM)

To add these fractions, our initial key step is to identify the Least Common Multiple (LCM) of the denominators involved: 11, 3, and 4. The LCM is the smallest multiple that these three numbers share, acting as a common ground for fraction operations. To find this, we can list the multiples of each number or use prime factorization.

Let's start by listing the multiples:

  • Multiples of 11: 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132...
  • Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132...
  • Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132...

Alternatively, we can use prime factorization. Here’s how it works:

  • Prime factorization of 11: 11 (11 is a prime number)
  • Prime factorization of 3: 3 (3 is a prime number)
  • Prime factorization of 4: 2 x 2 = 2²

The LCM is found by taking the highest power of each prime factor present in the numbers: 11 x 3 x 2² = 11 x 3 x 4 = 132. Thus, the least common multiple of 11, 3, and 4 is 132. This number will serve as our common denominator, enabling us to add the fractions together seamlessly. Using the LCM ensures that we are working with the smallest possible numbers, which simplifies our calculations and reduces the risk of errors.

Converting Fractions to a Common Denominator

Now that we've established the LCM as 132, the next crucial step is to convert each fraction to an equivalent form with this common denominator. This process involves adjusting both the numerator and the denominator of each fraction proportionally, ensuring the value of the fraction remains unchanged. This step is vital because fractions can only be added or subtracted when they share a common denominator, a principle rooted in the basic rules of arithmetic. Without this conversion, directly adding the numerators would be akin to comparing apples and oranges, leading to inaccurate results.

Let’s systematically convert each fraction:

  1. -9/11: To convert -9/11 to an equivalent fraction with a denominator of 132, we need to determine what number to multiply 11 by to get 132. By dividing 132 by 11, we find the factor to be 12. Therefore, we multiply both the numerator (-9) and the denominator (11) by 12:(-9 * 12) / (11 * 12) = -108/132
  2. 2/3: Similarly, for 2/3, we need to find the factor that, when multiplied by 3, gives us 132. Dividing 132 by 3 yields 44. Thus, we multiply both the numerator (2) and the denominator (3) by 44:(2 * 44) / (3 * 44) = 88/132
  3. -3/4: Lastly, for -3/4, we divide 132 by 4 to find the factor, which is 33. We then multiply both the numerator (-3) and the denominator (4) by 33:(-3 * 33) / (4 * 33) = -99/132

With these conversions, we now have three fractions that are ready to be added together: -108/132, 88/132, and -99/132. This process ensures that we are working with comparable quantities, paving the way for accurate addition. The significance of maintaining the fractions' original value cannot be overstated; it's a cornerstone of fractional arithmetic. By proportionally adjusting the numerators and denominators, we uphold this principle, setting the stage for the final calculations.

Adding the Fractions

Having successfully converted our fractions to a common denominator of 132, we are now poised to add them together. This step is where the fruits of our previous labor come to bear, as adding fractions with a common denominator is a straightforward process. The rule is simple: add the numerators while keeping the denominator the same. This rule is a direct consequence of the fundamental properties of fractions, which dictate how they can be combined. By adhering to this principle, we ensure that our final answer accurately represents the sum of the original fractions.

Let’s proceed with the addition:

-108/132 + 88/132 + (-99/132)

To make the calculation clearer, we can group the numerators together:

(-108 + 88 - 99) / 132

Now, let’s perform the arithmetic operations on the numerators:

First, add -108 and 88:

-108 + 88 = -20

Next, subtract 99 from -20:

-20 - 99 = -119

So, the sum of the numerators is -119. This gives us the fraction:

-119/132

This fraction represents the sum of the original three fractions. It's a single fraction that encapsulates the combined value of -9/11, 2/3, and -3/4. While this result is mathematically correct, it's often useful to consider whether the fraction can be simplified further. Simplifying a fraction means reducing it to its lowest terms, where the numerator and denominator have no common factors other than 1. This process enhances the clarity and utility of the answer, making it easier to compare and use in subsequent calculations.

Simplifying the Result

Our current result is -119/132. To simplify this fraction, we need to determine if the numerator (-119) and the denominator (132) share any common factors other than 1. If they do, we can divide both by their greatest common divisor (GCD) to reduce the fraction to its simplest form. This process of simplification is crucial for presenting the answer in its most concise and understandable form. It also aids in comparing the fraction with others and in performing further calculations with it. Let's delve into the process of finding the GCD.

To find the GCD, we can use several methods, such as listing factors or using the Euclidean algorithm. For simplicity, let's list the factors of 119 and 132:

  • Factors of 119: 1, 7, 17, 119
  • Factors of 132: 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132

Upon inspection, we find that the only common factor between 119 and 132 is 1. This means that the fraction -119/132 is already in its simplest form. There are no further reductions possible, as the numerator and denominator are relatively prime, sharing no common divisors other than unity. Consequently, our final answer remains as -119/132, a testament to the accuracy of our calculations and the inherent simplicity of the fraction in question.

Final Answer

In conclusion, after meticulously following each step—finding the LCM, converting fractions to a common denominator, adding the numerators, and simplifying the result—we arrive at the final answer. The sum of -9/11, 2/3, and -3/4 is:

-119/132

This fraction is in its simplest form, representing the precise sum of the given fractions. Understanding and mastering these steps is crucial for anyone looking to excel in mathematics, as they form the foundation for more complex calculations involving fractions. The ability to confidently add and subtract fractions is not just a mathematical skill; it's a problem-solving tool that extends to various aspects of life. From cooking to construction, fractions are integral to many real-world applications, making this a valuable skill to possess. Therefore, practice and understanding of these principles are highly recommended for mathematical proficiency.