Solving 1/9 × 9 - (2 × 1/3) - 1/3 A Step-by-Step Guide

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In the realm of mathematics, precision and order of operations are paramount. This article delves into the step-by-step solution of a seemingly simple yet potentially tricky equation: 1/9 × 9 - (2 × 1/3) - 1/3. We will meticulously break down each step, ensuring clarity and understanding for anyone seeking to master the fundamentals of arithmetic.

The Order of Operations: A Guiding Principle

Before we even begin to manipulate the numbers, it's crucial to acknowledge the hierarchy of mathematical operations. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), this principle dictates the sequence in which we perform calculations. It ensures consistency and accuracy in our results.

Deciphering the Equation: A Step-by-Step Approach

Our equation, 1/9 × 9 - (2 × 1/3) - 1/3, presents a combination of fractions, multiplication, subtraction, and parentheses. To conquer it effectively, we must adhere to PEMDAS religiously.

  1. Parentheses First: The presence of parentheses signals the initial step. We must resolve the expression within the parentheses before proceeding further. In our case, we have (2 × 1/3). Multiplying 2 by 1/3 yields 2/3. This simplifies our equation to 1/9 × 9 - 2/3 - 1/3.

  2. Multiplication Next: With the parentheses resolved, we turn our attention to multiplication. We have 1/9 × 9. A fraction multiplied by its reciprocal equals 1. Therefore, 1/9 × 9 = 1. Our equation now transforms into 1 - 2/3 - 1/3.

  3. Subtraction from Left to Right: Now, we encounter a series of subtractions. According to PEMDAS, addition and subtraction are performed from left to right. Thus, we first address 1 - 2/3. To subtract fractions, we need a common denominator. We can rewrite 1 as 3/3. Therefore, 3/3 - 2/3 = 1/3. Our equation is now streamlined to 1/3 - 1/3.

  4. The Final Subtraction: Finally, we perform the last subtraction: 1/3 - 1/3. This results in 0. Therefore, the final answer to our equation is 0.

The Solution Unveiled: 1/9 × 9 - (2 × 1/3) - 1/3 = 0

Through a meticulous application of the order of operations, we have successfully navigated the intricacies of the equation. The final answer, 0, highlights the importance of precision and adherence to mathematical principles. This exercise not only provides a solution but also underscores the elegance and logic inherent in mathematical problem-solving.

Exploring the Nuances of Fraction Arithmetic

At the heart of this equation lies the manipulation of fractions. Fractions, representing parts of a whole, often pose a challenge to those unfamiliar with their properties. Let's delve deeper into the fundamental operations involving fractions.

Mastering Multiplication of Fractions

The multiplication of fractions is arguably the simplest operation. To multiply two fractions, we simply multiply the numerators (the top numbers) and the denominators (the bottom numbers). For instance, (a/b) × (c/d) = (a × c) / (b × d). In our equation, we encountered the multiplication 2 × 1/3. We can consider 2 as the fraction 2/1. Applying the rule, we get (2/1) × (1/3) = (2 × 1) / (1 × 3) = 2/3.

Unveiling the Subtraction of Fractions

Subtracting fractions requires a common denominator. The common denominator is the least common multiple (LCM) of the denominators of the fractions involved. Once the fractions share a common denominator, we subtract the numerators while keeping the denominator constant. For example, to subtract 1/3 from 2/3, we already have a common denominator of 3. Therefore, 2/3 - 1/3 = (2 - 1) / 3 = 1/3. In our equation, we encountered the subtraction 1 - 2/3. To perform this, we rewrote 1 as 3/3, giving us 3/3 - 2/3 = 1/3.

The Significance of the Reciprocal

In the equation, we encountered 1/9 × 9. This showcases the concept of a reciprocal. The reciprocal of a fraction a/b is b/a. When a fraction is multiplied by its reciprocal, the result is always 1. In our case, 9 can be considered as the fraction 9/1, and its reciprocal is 1/9. Therefore, (9/1) × (1/9) = 1.

The Answer and Its Implications

The final answer to the equation 1/9 × 9 - (2 × 1/3) - 1/3 is 0. This seemingly simple result carries significant implications. It demonstrates the delicate balance within mathematical expressions and the power of order of operations. A slight deviation from the correct sequence can lead to a completely different outcome.

The answer also reinforces the concept of additive inverses. The terms 2/3 and 1/3, when combined and subtracted from 1 (which resulted from 1/9 × 9), effectively cancel each other out, leading to zero. This showcases the fundamental principle that every number has an additive inverse, and their sum is always zero.

Why This Matters: Practical Applications of Order of Operations

The order of operations, as demonstrated in this equation, is not just an abstract mathematical concept; it has profound practical applications in various fields.

Computer Programming

In computer programming, the order of operations is crucial for writing correct code. Programming languages follow specific rules for evaluating expressions, and a misunderstanding of these rules can lead to bugs and unexpected behavior. For instance, if a programmer intends to calculate (a + b) × c but writes a + b × c, the result will be different due to the precedence of multiplication over addition. Adhering to the order of operations ensures that the code executes as intended.

Spreadsheets and Data Analysis

Spreadsheets, widely used for data analysis and financial modeling, rely heavily on the order of operations. Formulas in spreadsheets often involve complex calculations, and the correct sequence of operations is vital for obtaining accurate results. For example, calculating compound interest requires a specific order of operations to ensure the correct amount is computed.

Engineering and Physics

In engineering and physics, formulas often involve multiple operations, including exponents, trigonometric functions, and more. Engineers and physicists must be meticulous in applying the order of operations to ensure the accuracy of their calculations. Errors in these calculations can have significant consequences in real-world applications.

Everyday Life

Even in everyday life, the order of operations comes into play. For example, when calculating the total cost of items with discounts and taxes, we implicitly follow the order of operations. We first calculate the discount, then add the tax, ensuring we arrive at the correct final price.

Conclusion: The Beauty of Mathematical Precision

The journey through the equation 1/9 × 9 - (2 × 1/3) - 1/3 has been more than just a mathematical exercise. It has been an exploration of fundamental principles, a testament to the power of order of operations, and a demonstration of the elegance inherent in mathematics. The final answer, 0, serves as a reminder that precision and adherence to rules are paramount in the world of numbers. By understanding and applying these principles, we unlock the ability to solve complex problems and navigate the mathematical landscape with confidence. This seemingly simple equation, therefore, becomes a gateway to a deeper appreciation of the beauty and utility of mathematics in all aspects of life.

This exploration highlights the importance of not only arriving at the correct answer but also understanding the underlying concepts and principles. The ability to break down a problem into smaller steps, apply the appropriate rules, and interpret the results is a valuable skill that extends far beyond the realm of mathematics. It fosters critical thinking, problem-solving abilities, and a deeper understanding of the world around us.