Evaluating Expressions Using Order Of Operations A Step By Step Guide
Evaluating mathematical expressions requires a clear understanding of the order of operations. This principle ensures that we arrive at the correct answer by performing calculations in a standardized sequence. The commonly used acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) helps us remember this order. In this article, we will dissect the expression $\frac{1}{3} \cdot \frac{3}{9}-\frac{1}{6} \cdot \frac{2}{6}$ step-by-step, applying the rules of order of operations to arrive at the correct solution. Mastering this concept is crucial for success in algebra and beyond, laying the foundation for more complex mathematical problem-solving.
Breaking Down the Expression
To accurately evaluate the given expression, $ rac{1}{3} \cdot \frac{3}{9}-\frac{1}{6} \cdot \frac{2}{6}$, we will follow the PEMDAS rule meticulously. This acronym guides us through the steps: Parentheses, Exponents, Multiplication and Division (from left to right), and finally, Addition and Subtraction (from left to right). Our expression involves multiplication and subtraction, so we will focus on these operations in the correct order. Before we dive into the arithmetic, it’s beneficial to understand the structure of the expression and identify the operations we need to perform. The expression consists of two multiplication operations separated by a subtraction. This means we'll first handle the multiplications and then the subtraction. Approaching the problem in this systematic way helps to minimize errors and ensures a clear and logical solution process. Understanding the structure is the first step towards mastering the order of operations and tackling more complex mathematical problems.
Step 1: Multiplication Operations
The first step in evaluating the expression $ rac1}{3} \cdot \frac{3}{9}-\frac{1}{6} \cdot \frac{2}{6}$ involves performing the multiplication operations. According to the order of operations (PEMDAS), multiplication and division are performed before addition and subtraction. We have two multiplication operations in this expression{3} \cdot \frac{3}{9}$ and $ rac{1}{6} \cdot \frac{2}{6}$. Let’s tackle them one at a time. To multiply fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. For the first multiplication, $ rac{1}{3} \cdot \frac{3}{9}$, we multiply 1 by 3 to get the new numerator and 3 by 9 to get the new denominator. This gives us $ rac{3}{27}$. Similarly, for the second multiplication, $ rac{1}{6} \cdot \frac{2}{6}$, we multiply 1 by 2 to get the numerator and 6 by 6 to get the denominator, resulting in $ rac{2}{36}$. By following this process, we've successfully completed the multiplication steps, which sets the stage for the next operation in our expression.
Step 2: Simplifying the Fractions
After performing the multiplication operations in the expression $\frac{1}{3} \cdot \frac{3}{9}-\frac{1}{6} \cdot \frac{2}{6}$, we arrived at the fractions $\frac{3}{27}$ and $ rac{2}{36}$. Simplifying fractions is a crucial step in mathematics as it makes the numbers easier to work with and helps to present the final answer in its simplest form. A fraction is simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD). Let's start with the fraction $\frac{3}{27}$. The GCD of 3 and 27 is 3. Dividing both the numerator and the denominator by 3, we get $\frac{3 \div 3}{27 \div 3} = \frac{1}{9}$. Now, let's simplify the fraction $\frac{2}{36}$. The GCD of 2 and 36 is 2. Dividing both the numerator and the denominator by 2, we get $\frac{2 \div 2}{36 \div 2} = \frac{1}{18}$. By simplifying these fractions, we have transformed our expression into $\frac{1}{9} - \frac{1}{18}$, which is much easier to handle in the subsequent subtraction step. Simplifying fractions is not just a matter of reducing numbers; it's a fundamental skill that enhances mathematical clarity and efficiency.
Step 3: Subtraction Operation
Having simplified the fractions, our expression now reads $ rac{1}{9} - \frac{1}{18}$. The next step, according to the order of operations, is to perform the subtraction. To subtract fractions, they must have a common denominator. In this case, the denominators are 9 and 18. The least common multiple (LCM) of 9 and 18 is 18, which will serve as our common denominator. We need to convert $ rac{1}{9}$ into an equivalent fraction with a denominator of 18. To do this, we multiply both the numerator and the denominator of $ rac{1}{9}$ by 2, resulting in $\frac{1 \cdot 2}{9 \cdot 2} = \frac{2}{18}$. Now our expression becomes $\frac{2}{18} - \frac{1}{18}$. With a common denominator, we can subtract the numerators directly. Subtracting 1 from 2 gives us 1. So, the result of the subtraction is $\frac{1}{18}$. This completes the final operation in our expression, leading us to the solution. Understanding how to find a common denominator and perform fraction subtraction is a key skill in mathematics, essential for solving a wide range of problems.
The Final Result
After meticulously following the order of operations, we have successfully evaluated the expression $ rac{1}{3} \cdot \frac{3}{9}-\frac{1}{6} \cdot \frac{2}{6}$. We began by performing the multiplication operations, which resulted in $ rac{3}{27}$ and $ rac{2}{36}$. We then simplified these fractions to $ rac{1}{9}$ and $ rac{1}{18}$, respectively. Finally, we subtracted the simplified fractions, finding a common denominator and performing the subtraction, which led us to the final answer of $\frac{1}{18}$. This step-by-step process highlights the importance of adhering to the order of operations to arrive at the correct solution. The result, $\frac{1}{18}$, represents the simplified value of the original expression, showcasing the power of mathematical operations in reducing complexity. This exercise not only provides a numerical answer but also reinforces the principles of arithmetic and algebraic manipulation, which are foundational for advanced mathematical studies. Understanding and applying these principles ensures accuracy and efficiency in problem-solving.
Conclusion
In conclusion, evaluating the expression $ rac{1}{3} \cdot \frac{3}{9}-\frac{1}{6} \cdot \frac{2}{6}$ has been a comprehensive exercise in applying the order of operations. By systematically working through the expression, we've reinforced the importance of PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) in mathematical calculations. We began with multiplication, simplified the resulting fractions, and then performed subtraction, ultimately arriving at the solution $ rac{1}{18}$. This process not only demonstrates the correct methodology for solving such expressions but also underscores the significance of simplifying fractions and finding common denominators. The journey from the initial expression to the final result is a testament to the power of structured problem-solving in mathematics. Mastering the order of operations is crucial for students and professionals alike, as it forms the bedrock of more complex mathematical concepts and applications. This example serves as a valuable reference for anyone seeking to enhance their mathematical skills and understanding.
